{"id":5445,"date":"2023-06-29T17:40:07","date_gmt":"2023-06-29T17:40:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=5445"},"modified":"2024-10-18T20:50:40","modified_gmt":"2024-10-18T20:50:40","slug":"integers-learn-it-5","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/integers-learn-it-5\/","title":{"raw":"Integers: Learn It 5","rendered":"Integers: Learn It 5"},"content":{"raw":"<h2>Subtracting Integers<\/h2>\r\n<p>Building on the fundamental principles of addition, we now turn our attention to the operation of subtraction. Subtraction often requires us to think in reverse. <br \/>\r\n<br \/>\r\nJust as with addition, we can use counters to help us track the difference between positive and negative values in this process. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.<\/p>\r\n<section class=\"textbox example\">\r\n<p>We will model four subtraction scenarios\u00a0using the numbers [latex]5[\/latex] and [latex]3[\/latex].<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex]5 - 3[\/latex]<\/li>\r\n\t<li>[latex]- 5-\\left(-3\\right)[\/latex]<\/li>\r\n\t<li>[latex]-5 - 3[\/latex]<\/li>\r\n\t<li>[latex]5-\\left(-3\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>\r\n<table id=\"eip-id1168469608926\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus 3 means 5 take away 3. The second row states model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so take away 3 positives. It is followed by 5 blue circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 blue circles that are labeled two positives. The fifth row states the difference of 5 and 3 is 2. It is followed by the expression 5 minus 3 equals 2.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]5 - 3[\/latex] means [latex]5[\/latex] take away [latex]3[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15509\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215401\/SubtractingIntegers_1.png\" alt=\"This figure has 5 blue circles with the number 5 underneath.\" width=\"354\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away the second number. So take away [latex]3[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15510\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215411\/SubtractingIntegers_2.png\" alt=\"This figure has 5 blue circles. Under the last two circles, it say 2 positives.\" width=\"356\" height=\"96\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the counters that are left.<\/td>\r\n<td><img class=\"alignnone wp-image-15511\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215420\/SubtractingIntegers_3.png\" alt=\"This figure has two blue circles.\" width=\"359\" height=\"66\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>\r\n<p>[latex]5 - 3=2[\/latex]<\/p>\r\n<p>The difference between [latex]5[\/latex] and [latex]3[\/latex] is [latex]2[\/latex].<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table id=\"eip-id1168469693755\" class=\"unnumbered unstyled\" style=\"width: 75%; height: 378px;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so take away 3 negatives. It is followed by 5 red circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 red circles which are labeled two negatives. The fifth row states the difference of negative 5 and negative 3 is negative 2. Next to this is the expression negative 5 minus negative 3 equals negative two.\">\r\n<tbody>\r\n<tr style=\"height: 55px;\">\r\n<td style=\"height: 55px;\">Interpret the expression.<\/td>\r\n<td style=\"height: 55px;\">[latex]-5-\\left(-3\\right)[\/latex] means [latex]-5[\/latex] take away [latex]-3[\/latex].<\/td>\r\n<\/tr>\r\n<tr style=\"height: 62px;\">\r\n<td style=\"height: 62px;\">Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\r\n<td style=\"height: 62px;\"><img class=\"wp-image-15514 alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215454\/SubtractingIntegers_6.png\" alt=\"This figure has 5 red circles labeled with a negative 5\" width=\"356\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 70px;\">\r\n<td style=\"height: 70px;\">Take away the second number. So take away [latex]3[\/latex] negatives.<\/td>\r\n<td style=\"height: 70px;\"><img class=\"alignnone wp-image-15515\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215504\/SubtractingIntegers_7.png\" alt=\"This figure has 5 red circles. Three are circled. Underneath the two that are not it is labeled 2 negatives.\" width=\"354\" height=\"95\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 49px;\">\r\n<td style=\"height: 49px;\">Find the number of counters that are left.<\/td>\r\n<td style=\"height: 49px;\"><img class=\"alignnone wp-image-15516\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215514\/SubtractingIntegers_8.png\" alt=\"This figure has two red circles.\" width=\"357\" height=\"66\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 142px;\">\r\n<td style=\"height: 142px;\">\u00a0<\/td>\r\n<td style=\"height: 142px;\">\r\n<p>[latex]-5-\\left(-3\\right)=-2[\/latex]<\/p>\r\n<p>The difference between [latex]-5[\/latex] and [latex]-3[\/latex] is [latex]-2[\/latex].<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table id=\"eip-id1168468477542\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by negative 5 minus 3 means negative 5 take away 3. The second row states to model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so we need to take away 3 positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The fourth row states there are no positives to take away so add neutral pairs until you have 3 positives. The fifth row states now take away three positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The blue shaded circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 red circles. The seventh row states the difference of negative five and three is negative 8. It is followed by the expression negative 5 minus 3 equals negative 8.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]-5 - 3[\/latex] means [latex]-5[\/latex] take away [latex]3[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15520\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215754\/SubtractingIntegers_13.png\" alt=\"This figure has 5 red circles labeled negative 5.\" width=\"355\" height=\"77\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Take away the second number.<\/p>\r\n<p>So we need to take away [latex]3[\/latex] positives.<\/p>\r\n<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>But there are no positives to take away.<\/p>\r\n<p>Add neutral pairs until you have [latex]3[\/latex] positives.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone wp-image-15521\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215806\/SubtractingIntegers_14.png\" alt=\"This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles.\" width=\"357\" height=\"99\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Now take away [latex]3[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15522\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215816\/SubtractingIntegers_15.png\" alt=\"This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles. The 3 blue circles are circled together.\" width=\"355\" height=\"131\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of counters that are left.<\/td>\r\n<td><img class=\"alignnone wp-image-15523\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215827\/SubtractingIntegers_16.png\" alt=\"This figure has 8 red circles labeled 8 negatives.\" width=\"354\" height=\"76\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>\r\n<p>[latex]-5 - 3=-8[\/latex]<\/p>\r\n<p>The difference of [latex]-5[\/latex] and [latex]3[\/latex] is [latex]-8[\/latex].<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table id=\"eip-id1168467234371\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states to model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so we need to take away 3 negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The fourth row states there are no negatives to take away so add neutral pairs until you have 3 negatives. The fifth row states now take away three negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The red circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 blue circles. The seventh row states the difference of 5 and negative 3 is eight, beside it the expression 5 minus negative 3 equals eight.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]5-\\left(-3\\right)[\/latex] means [latex]5[\/latex] take away [latex]-3[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15526\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215905\/SubtractingIntegers_19.png\" alt=\"This figure has 5 blue circles with the number 5 labeled underneath.\" width=\"357\" height=\"77\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away the second number, so take away [latex]3[\/latex] negatives.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>But there are no negatives to take away.<\/p>\r\n<p>Add neutral pairs until you have [latex]3[\/latex] negatives.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone wp-image-15527\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220212\/SubtractingIntegers_20.png\" alt=\"This figure has 8 blue circles with a space between the first 5 and last 3. Underneath the last 3 blue circles there are 3 red circles.\" width=\"357\" height=\"99\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15528\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220223\/SubtractingIntegers_21.png\" alt=\"This figure has 8 blue circles with a space between the first 5 and last 3. Underneath the last 3 blue circles there are 3 red circles that are circled together. \" width=\"357\" height=\"132\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of counters that are left.<\/td>\r\n<td><img class=\"alignnone wp-image-15529\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220233\/SubtractingIntegers_22.png\" alt=\"8 blue circles labeled 8 positives.\" width=\"357\" height=\"77\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>\r\n<p>The difference of [latex]5[\/latex] and [latex]-3[\/latex] is [latex]8[\/latex].<\/p>\r\n<p>[latex]5-\\left(-3\\right)=8[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p>Now you can try an example that summarizes the situations above, with different numbers. Recall the different scenarios:<\/p>\r\n<ul>\r\n\t<li>subtracting a positive number from a positive number<\/li>\r\n\t<li>subtracting a positive number from a negative number<\/li>\r\n\t<li>subtracting a negative number from a positive number<\/li>\r\n\t<li>subtracting a negative number from a negative number<\/li>\r\n<\/ul>\r\n<section class=\"textbox example\">Model each subtraction.\r\n\r\n<ol>\r\n\t<li>[latex]8 \u2212 2[\/latex]<\/li>\r\n\t<li>[latex]\u22125 \u2212 4[\/latex]<\/li>\r\n\t<li>[latex]6 \u2212 (\u22126)[\/latex]<\/li>\r\n\t<li>[latex]\u22128 \u2212 (\u22123)[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"214895\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"214895\"]<\/p>\r\n<ol>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>\r\n<p>[latex]8 - 2[\/latex]<\/p>\r\n<p>This means [latex]8[\/latex] take away [latex]2[\/latex].<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]8[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15532\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220310\/SubtractingIntegers_25.png\" alt=\"8 blue circles.\" width=\"355\" height=\"63\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]2[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15533\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220320\/SubtractingIntegers_26.png\" alt=\"8 blue circles with a space between the first 6 and last 2. The last 2 circles are circled together. \" width=\"355\" height=\"63\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]8 - 2=6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>\r\n<p>[latex]-5 - 4[\/latex]<\/p>\r\n<p>This means [latex]-5[\/latex] take away [latex]4[\/latex].<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]5[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15534\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220333\/SubtractingIntegers_27.png\" alt=\"5 red circles.\" width=\"354\" height=\"63\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>You need to take away [latex]4[\/latex] positives.<\/p>\r\n<p>Add [latex]4[\/latex] neutral pairs to get [latex]4[\/latex] positives.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone wp-image-15535\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220349\/SubtractingIntegers_28.png\" alt=\"9 circles with a space between the first 5 and last 4. Underneath the last 4 red circles there are 4 blue circles.\" width=\"357\" height=\"102\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]4[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15536\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220401\/SubtractingIntegers_29.png\" alt=\"9 red circles with a space between the first 5 and last 4. Underneath the last 4 red circles there are 4 blue circles that are circled together.\" width=\"355\" height=\"122\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignnone wp-image-15537\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220411\/SubtractingIntegers_30.png\" alt=\"9 red circles.\" width=\"354\" height=\"63\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]-5 - 4=-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>\r\n<p>[latex]6-\\left(-6\\right)[\/latex]<\/p>\r\n<p>This means [latex]6[\/latex] take away [latex]-6[\/latex].<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]6[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15538\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220422\/SubtractingIntegers_31.png\" alt=\"6 blue circles.\" width=\"399\" height=\"53\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]6[\/latex] neutrals to get [latex]6[\/latex] negatives to take away.<\/td>\r\n<td><img class=\"alignnone wp-image-15539\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220433\/SubtractingIntegers_32.png\" alt=\"12 blue circles with a space between the first 6 and the last 6. Underneath the last 6 blue circles, there are 6 red circles.\" width=\"399\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove [latex]6[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15540\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220444\/SubtractingIntegers_33.png\" alt=\"12 blue circles with a space between the first 6 and the last 6. Underneath the last 6 blue circles, there are 6 red circles that are circled together.\" width=\"406\" height=\"105\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignnone wp-image-15541\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220501\/SubtractingIntegers_34.png\" alt=\"12 blue circles.\" width=\"405\" height=\"54\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]6-\\left(-6\\right)=12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>\r\n<p>[latex]-8-\\left(-3\\right)[\/latex]<\/p>\r\n<p>This means [latex]-8[\/latex] take away [latex]-3[\/latex].<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]8[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15542\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220514\/SubtractingIntegers_35.png\" alt=\"8 red circles\" width=\"355\" height=\"65\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15543\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220524\/SubtractingIntegers_36.png\" alt=\"8 red circles with a space between the first five and last three. The last three circles are circled together.\" width=\"357\" height=\"69\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignnone wp-image-15534\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220333\/SubtractingIntegers_27.png\" alt=\"5 red circles\" width=\"352\" height=\"63\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]-8-\\left(-3\\right)=-5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Each of the examples so far have been carefully constructed so that the sign of the answer matched the sign of the first number in the expression. \u00a0For example, in\u00a0\u00a0[latex]\u22125 \u2212 4[\/latex], the result is [latex]-9[\/latex], which is the same sign as [latex]-5[\/latex]. Now we will see subtraction where the sign of the result is different from the starting number.<\/p>\r\n<section class=\"textbox example\">Model each subtraction expression:\r\n\r\n<ol>\r\n\t<li>[latex]2 - 8[\/latex]<\/li>\r\n\t<li>[latex]-3-\\left(-8\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"214815\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"214815\"]<\/p>\r\n<ol>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>We start with [latex]2[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15552\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220734\/SubtractingIntegers_45.png\" alt=\"2 blue circles with a number 2 labeled underneath.\" width=\"356\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We need to take away [latex]8[\/latex] positives, but we have only [latex]2[\/latex].<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add neutral pairs until there are [latex]8[\/latex] positives to take away.<\/td>\r\n<td><img class=\"alignnone wp-image-15553\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220747\/SubtractingIntegers_46.png\" alt=\"8 blue circles with a space between the first 2 and last 6. Underneath the last 6 blue circles there 6 red circles.\" width=\"356\" height=\"93\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away [latex]8[\/latex]\u00a0positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15554\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220758\/SubtractingIntegers_47.png\" alt=\"8 blue circles that are circled together. Underneath there are 6 red circles.\" width=\"354\" height=\"138\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Find the number of counters that are left.<\/p>\r\n<p>There are [latex]6[\/latex] negatives.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone wp-image-15555\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220809\/SubtractingIntegers_48.png\" alt=\"6 red circles labeled 6 negatives\" width=\"356\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]2 - 8=-6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>We start with [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15556\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220821\/SubtractingIntegers_49.png\" alt=\"3 red circles with a number 3 underneath\" width=\"354\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We need to take away [latex]8[\/latex] negatives, but we have only [latex]3[\/latex].<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add neutral pairs until there are [latex]8[\/latex] negatives to take away.<\/td>\r\n<td><img class=\"alignnone wp-image-15557\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220831\/SubtractingIntegers_50.png\" alt=\"7 red circles and below there are 6 blue circles\" width=\"358\" height=\"94\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away the [latex]8[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15558\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220843\/SubtractingIntegers_51.png\" alt=\"8 red circles that are circled together. Below there are 5 blue circles\" width=\"358\" height=\"139\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Find the number of counters that are left.<\/p>\r\n<p>There are [latex]5[\/latex] positives.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone wp-image-15559\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220856\/SubtractingIntegers_52.png\" alt=\"5 blue circles labeled 5 positives\" width=\"358\" height=\"76\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]-3-\\left(-8\\right)=5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Now that you have seen subtraction modeled with color counters,\u00a0we can move on to performing subtraction of integers without the models.<\/p>\r\n<section class=\"textbox example\">\r\n<ul id=\"fs-id2376804\">\r\n\t<li>Subtract [latex]-23 - 7[\/latex].<br \/>\r\n<ul id=\"fs-id2376804\">\r\n\t<li>Think: We start with [latex]23[\/latex] negative counters. We have to subtract [latex]7[\/latex] positives, but there are no positives to take away. So we add [latex]7[\/latex] neutral pairs to get the [latex]7[\/latex] positives. Now we take away the [latex]7[\/latex] positives. So what\u2019s left? We have the original [latex]23[\/latex] negatives plus [latex]7[\/latex] more negatives from the neutral pair. The result is [latex]30[\/latex] negatives.<center>[latex]-23 - 7=-30[\/latex]<br \/>\r\n<br \/>\r\n<\/center>Notice, that to subtract [latex]\\text{7,}[\/latex] we added [latex]7[\/latex] negatives.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Subtract [latex]30-\\left(-12\\right)[\/latex].<br \/>\r\n<ul id=\"fs-id2376804\">\r\n\t<li>Think: We start with [latex]30[\/latex] positives. We have to subtract [latex]12[\/latex] negatives, but there are no negatives to take away. So we add [latex]12[\/latex] neutral pairs to the [latex]30[\/latex] positives. Now we take away the [latex]12[\/latex] negatives. What\u2019s left? We have the original [latex]30[\/latex] positives plus [latex]12[\/latex] more positives from the neutral pairs. The result is [latex]42[\/latex] positives.<center>[latex]30-\\left(-12\\right)=42[\/latex]<br \/>\r\n<br \/>\r\n<\/center>Notice that to subtract [latex]-12[\/latex], we added [latex]12[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.<\/p>\r\n<section class=\"textbox proTip\">Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows: Subtracting a number is the same as adding its opposite.<center>[latex]a-b=a+(-b)[\/latex]<\/center><\/section>\r\n<p>Now let\u2019s increase the complexity of the examples a little bit. We will use the order of operations to simplify terms in parentheses before we subtract from left to right.<\/p>\r\n<section class=\"textbox example\">Simplify: [latex]7-\\left(-4 - 3\\right)-9[\/latex]<br \/>\r\n[reveal-answer q=\"769817\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"769817\"]<br \/>\r\nWe use the order of operations to simplify this expression, performing operations inside the parentheses first. Then we subtract from left to right.\r\n\r\n<table id=\"eip-id1168469609684\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Simplify inside the parentheses first.<\/td>\r\n<td>[latex]7-\\left(-4 - 3\\right)-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract from left to right.<\/td>\r\n<td>[latex]7-\\left(-7\\right)-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]14-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8904[\/ohm2_question]<\/section>\r\n<h2>Applications With Subtracting Integers<\/h2>\r\n<p>It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it. So when we solve an application problem, we first need to determine what we are asked to find. Then we can write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.<\/p>\r\n<p>Geography provides an application of negative numbers with the elevations of places below sea level.<\/p>\r\n<section class=\"textbox example\">Dinesh hiked from Mt. Whitney, the highest point in California, to Death Valley, the lowest point. The elevation of Mt. Whitney is [latex]14,497[\/latex] feet above sea level and the elevation of Death Valley is [latex]282[\/latex] feet below sea level. What is the difference in elevation between Mt. Whitney and Death Valley?<br \/>\r\n[reveal-answer q=\"128798\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"128798\"]\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 1. What are we asked to find?<\/td>\r\n<td>The difference in elevation between Mt. Whitney and Death Valley<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. Write a phrase.<\/td>\r\n<td>elevation of Mt. Whitney\u2212elevation of Death Valley<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. Translate.<\/td>\r\n<td>[latex]14,497-\\left(-282\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. Simplify.<\/td>\r\n<td>[latex]14,779[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. Write a complete sentence that answers the question.<\/td>\r\n<td>The difference in elevation is [latex]14,779[\/latex] feet.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8905[\/ohm2_question]<\/section>\r\n<p>Managing your money can involve both positive and negative numbers.<\/p>\r\n<section class=\"textbox example\">Leslie has [latex]$25[\/latex] in her checking account and she has a transaction for [latex]$8[\/latex].\r\n\r\n<ol>\r\n\t<li style=\"list-style-type: none;\">\r\n<ol>\r\n\t<li>What is the balance after the transaction?<\/li>\r\n\t<li>She has a second transaction for [latex]$20[\/latex]. What is the new balance after this transaction?<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"83739\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"83739\"]<\/p>\r\n<ol>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>What are we asked to find?<\/td>\r\n<td>The balance of the account<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write a phrase.<\/td>\r\n<td>[latex]$25[\/latex] minus [latex]$8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate<\/td>\r\n<td>[latex]$25-$8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]$17[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write a sentence answer.<\/td>\r\n<td>The balance is [latex]$17[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li><br \/>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>What are we asked to find?<\/td>\r\n<td>The new balance<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write a phrase.<\/td>\r\n<td>[latex]$17[\/latex] minus [latex]$20[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate<\/td>\r\n<td>[latex]$17-$20[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-$3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write a sentence answer.<\/td>\r\n<td>She is overdrawn by [latex]$3[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8906[\/ohm2_question]<\/section>","rendered":"<h2>Subtracting Integers<\/h2>\n<p>Building on the fundamental principles of addition, we now turn our attention to the operation of subtraction. Subtraction often requires us to think in reverse. <\/p>\n<p>Just as with addition, we can use counters to help us track the difference between positive and negative values in this process. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.<\/p>\n<section class=\"textbox example\">\n<p>We will model four subtraction scenarios\u00a0using the numbers [latex]5[\/latex] and [latex]3[\/latex].<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]5 - 3[\/latex]<\/li>\n<li>[latex]- 5-\\left(-3\\right)[\/latex]<\/li>\n<li>[latex]-5 - 3[\/latex]<\/li>\n<li>[latex]5-\\left(-3\\right)[\/latex]<\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\">\n<li>\n<table id=\"eip-id1168469608926\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus 3 means 5 take away 3. The second row states model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so take away 3 positives. It is followed by 5 blue circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 blue circles that are labeled two positives. The fifth row states the difference of 5 and 3 is 2. It is followed by the expression 5 minus 3 equals 2.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]5 - 3[\/latex] means [latex]5[\/latex] take away [latex]3[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15509\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215401\/SubtractingIntegers_1.png\" alt=\"This figure has 5 blue circles with the number 5 underneath.\" width=\"354\" height=\"85\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away the second number. So take away [latex]3[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15510\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215411\/SubtractingIntegers_2.png\" alt=\"This figure has 5 blue circles. Under the last two circles, it say 2 positives.\" width=\"356\" height=\"96\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the counters that are left.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15511\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215420\/SubtractingIntegers_3.png\" alt=\"This figure has two blue circles.\" width=\"359\" height=\"66\" \/><\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>\n[latex]5 - 3=2[\/latex]<\/p>\n<p>The difference between [latex]5[\/latex] and [latex]3[\/latex] is [latex]2[\/latex].<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table id=\"eip-id1168469693755\" class=\"unnumbered unstyled\" style=\"width: 75%; height: 378px;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so take away 3 negatives. It is followed by 5 red circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 red circles which are labeled two negatives. The fifth row states the difference of negative 5 and negative 3 is negative 2. Next to this is the expression negative 5 minus negative 3 equals negative two.\">\n<tbody>\n<tr style=\"height: 55px;\">\n<td style=\"height: 55px;\">Interpret the expression.<\/td>\n<td style=\"height: 55px;\">[latex]-5-\\left(-3\\right)[\/latex] means [latex]-5[\/latex] take away [latex]-3[\/latex].<\/td>\n<\/tr>\n<tr style=\"height: 62px;\">\n<td style=\"height: 62px;\">Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\n<td style=\"height: 62px;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-15514 alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215454\/SubtractingIntegers_6.png\" alt=\"This figure has 5 red circles labeled with a negative 5\" width=\"356\" height=\"85\" \/><\/td>\n<\/tr>\n<tr style=\"height: 70px;\">\n<td style=\"height: 70px;\">Take away the second number. So take away [latex]3[\/latex] negatives.<\/td>\n<td style=\"height: 70px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15515\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215504\/SubtractingIntegers_7.png\" alt=\"This figure has 5 red circles. Three are circled. Underneath the two that are not it is labeled 2 negatives.\" width=\"354\" height=\"95\" \/><\/td>\n<\/tr>\n<tr style=\"height: 49px;\">\n<td style=\"height: 49px;\">Find the number of counters that are left.<\/td>\n<td style=\"height: 49px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15516\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215514\/SubtractingIntegers_8.png\" alt=\"This figure has two red circles.\" width=\"357\" height=\"66\" \/><\/td>\n<\/tr>\n<tr style=\"height: 142px;\">\n<td style=\"height: 142px;\">\u00a0<\/td>\n<td style=\"height: 142px;\">\n[latex]-5-\\left(-3\\right)=-2[\/latex]<\/p>\n<p>The difference between [latex]-5[\/latex] and [latex]-3[\/latex] is [latex]-2[\/latex].<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table id=\"eip-id1168468477542\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by negative 5 minus 3 means negative 5 take away 3. The second row states to model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so we need to take away 3 positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The fourth row states there are no positives to take away so add neutral pairs until you have 3 positives. The fifth row states now take away three positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The blue shaded circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 red circles. The seventh row states the difference of negative five and three is negative 8. It is followed by the expression negative 5 minus 3 equals negative 8.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]-5 - 3[\/latex] means [latex]-5[\/latex] take away [latex]3[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15520\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215754\/SubtractingIntegers_13.png\" alt=\"This figure has 5 red circles labeled negative 5.\" width=\"355\" height=\"77\" \/><\/td>\n<\/tr>\n<tr>\n<td>\n<p>Take away the second number.<\/p>\n<p>So we need to take away [latex]3[\/latex] positives.<\/p>\n<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>\n<p>But there are no positives to take away.<\/p>\n<p>Add neutral pairs until you have [latex]3[\/latex] positives.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15521\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215806\/SubtractingIntegers_14.png\" alt=\"This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles.\" width=\"357\" height=\"99\" \/><\/td>\n<\/tr>\n<tr>\n<td>Now take away [latex]3[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15522\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215816\/SubtractingIntegers_15.png\" alt=\"This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles. The 3 blue circles are circled together.\" width=\"355\" height=\"131\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the number of counters that are left.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15523\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215827\/SubtractingIntegers_16.png\" alt=\"This figure has 8 red circles labeled 8 negatives.\" width=\"354\" height=\"76\" \/><\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>\n[latex]-5 - 3=-8[\/latex]<\/p>\n<p>The difference of [latex]-5[\/latex] and [latex]3[\/latex] is [latex]-8[\/latex].<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table id=\"eip-id1168467234371\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states to model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so we need to take away 3 negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The fourth row states there are no negatives to take away so add neutral pairs until you have 3 negatives. The fifth row states now take away three negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The red circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 blue circles. The seventh row states the difference of 5 and negative 3 is eight, beside it the expression 5 minus negative 3 equals eight.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]5-\\left(-3\\right)[\/latex] means [latex]5[\/latex] take away [latex]-3[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15526\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215905\/SubtractingIntegers_19.png\" alt=\"This figure has 5 blue circles with the number 5 labeled underneath.\" width=\"357\" height=\"77\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away the second number, so take away [latex]3[\/latex] negatives.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>\n<p>But there are no negatives to take away.<\/p>\n<p>Add neutral pairs until you have [latex]3[\/latex] negatives.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15527\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220212\/SubtractingIntegers_20.png\" alt=\"This figure has 8 blue circles with a space between the first 5 and last 3. Underneath the last 3 blue circles there are 3 red circles.\" width=\"357\" height=\"99\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15528\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220223\/SubtractingIntegers_21.png\" alt=\"This figure has 8 blue circles with a space between the first 5 and last 3. Underneath the last 3 blue circles there are 3 red circles that are circled together.\" width=\"357\" height=\"132\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the number of counters that are left.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15529\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220233\/SubtractingIntegers_22.png\" alt=\"8 blue circles labeled 8 positives.\" width=\"357\" height=\"77\" \/><\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>\n<p>The difference of [latex]5[\/latex] and [latex]-3[\/latex] is [latex]8[\/latex].<\/p>\n<p>[latex]5-\\left(-3\\right)=8[\/latex]\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/section>\n<p>Now you can try an example that summarizes the situations above, with different numbers. Recall the different scenarios:<\/p>\n<ul>\n<li>subtracting a positive number from a positive number<\/li>\n<li>subtracting a positive number from a negative number<\/li>\n<li>subtracting a negative number from a positive number<\/li>\n<li>subtracting a negative number from a negative number<\/li>\n<\/ul>\n<section class=\"textbox example\">Model each subtraction.<\/p>\n<ol>\n<li>[latex]8 \u2212 2[\/latex]<\/li>\n<li>[latex]\u22125 \u2212 4[\/latex]<\/li>\n<li>[latex]6 \u2212 (\u22126)[\/latex]<\/li>\n<li>[latex]\u22128 \u2212 (\u22123)[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214895\">Show Solution<\/button><\/p>\n<div id=\"q214895\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>\n[latex]8 - 2[\/latex]<\/p>\n<p>This means [latex]8[\/latex] take away [latex]2[\/latex].<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]8[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15532\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220310\/SubtractingIntegers_25.png\" alt=\"8 blue circles.\" width=\"355\" height=\"63\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]2[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15533\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220320\/SubtractingIntegers_26.png\" alt=\"8 blue circles with a space between the first 6 and last 2. The last 2 circles are circled together.\" width=\"355\" height=\"63\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td>[latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]8 - 2=6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>\n[latex]-5 - 4[\/latex]<\/p>\n<p>This means [latex]-5[\/latex] take away [latex]4[\/latex].<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]5[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15534\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220333\/SubtractingIntegers_27.png\" alt=\"5 red circles.\" width=\"354\" height=\"63\" \/><\/td>\n<\/tr>\n<tr>\n<td>\n<p>You need to take away [latex]4[\/latex] positives.<\/p>\n<p>Add [latex]4[\/latex] neutral pairs to get [latex]4[\/latex] positives.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15535\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220349\/SubtractingIntegers_28.png\" alt=\"9 circles with a space between the first 5 and last 4. Underneath the last 4 red circles there are 4 blue circles.\" width=\"357\" height=\"102\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]4[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15536\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220401\/SubtractingIntegers_29.png\" alt=\"9 red circles with a space between the first 5 and last 4. Underneath the last 4 red circles there are 4 blue circles that are circled together.\" width=\"355\" height=\"122\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15537\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220411\/SubtractingIntegers_30.png\" alt=\"9 red circles.\" width=\"354\" height=\"63\" \/><\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]-5 - 4=-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>\n[latex]6-\\left(-6\\right)[\/latex]<\/p>\n<p>This means [latex]6[\/latex] take away [latex]-6[\/latex].<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]6[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15538\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220422\/SubtractingIntegers_31.png\" alt=\"6 blue circles.\" width=\"399\" height=\"53\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]6[\/latex] neutrals to get [latex]6[\/latex] negatives to take away.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15539\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220433\/SubtractingIntegers_32.png\" alt=\"12 blue circles with a space between the first 6 and the last 6. Underneath the last 6 blue circles, there are 6 red circles.\" width=\"399\" height=\"85\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove [latex]6[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15540\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220444\/SubtractingIntegers_33.png\" alt=\"12 blue circles with a space between the first 6 and the last 6. Underneath the last 6 blue circles, there are 6 red circles that are circled together.\" width=\"406\" height=\"105\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15541\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220501\/SubtractingIntegers_34.png\" alt=\"12 blue circles.\" width=\"405\" height=\"54\" \/><\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]6-\\left(-6\\right)=12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>\n[latex]-8-\\left(-3\\right)[\/latex]<\/p>\n<p>This means [latex]-8[\/latex] take away [latex]-3[\/latex].<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]8[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15542\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220514\/SubtractingIntegers_35.png\" alt=\"8 red circles\" width=\"355\" height=\"65\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15543\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220524\/SubtractingIntegers_36.png\" alt=\"8 red circles with a space between the first five and last three. The last three circles are circled together.\" width=\"357\" height=\"69\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15534\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220333\/SubtractingIntegers_27.png\" alt=\"5 red circles\" width=\"352\" height=\"63\" \/><\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]-5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]-8-\\left(-3\\right)=-5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>Each of the examples so far have been carefully constructed so that the sign of the answer matched the sign of the first number in the expression. \u00a0For example, in\u00a0\u00a0[latex]\u22125 \u2212 4[\/latex], the result is [latex]-9[\/latex], which is the same sign as [latex]-5[\/latex]. Now we will see subtraction where the sign of the result is different from the starting number.<\/p>\n<section class=\"textbox example\">Model each subtraction expression:<\/p>\n<ol>\n<li>[latex]2 - 8[\/latex]<\/li>\n<li>[latex]-3-\\left(-8\\right)[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214815\">Show Solution<\/button><\/p>\n<div id=\"q214815\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td>We start with [latex]2[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15552\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220734\/SubtractingIntegers_45.png\" alt=\"2 blue circles with a number 2 labeled underneath.\" width=\"356\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>We need to take away [latex]8[\/latex] positives, but we have only [latex]2[\/latex].<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Add neutral pairs until there are [latex]8[\/latex] positives to take away.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15553\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220747\/SubtractingIntegers_46.png\" alt=\"8 blue circles with a space between the first 2 and last 6. Underneath the last 6 blue circles there 6 red circles.\" width=\"356\" height=\"93\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away [latex]8[\/latex]\u00a0positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15554\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220758\/SubtractingIntegers_47.png\" alt=\"8 blue circles that are circled together. Underneath there are 6 red circles.\" width=\"354\" height=\"138\" \/><\/td>\n<\/tr>\n<tr>\n<td>\n<p>Find the number of counters that are left.<\/p>\n<p>There are [latex]6[\/latex] negatives.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15555\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220809\/SubtractingIntegers_48.png\" alt=\"6 red circles labeled 6 negatives\" width=\"356\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]2 - 8=-6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>We start with [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15556\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220821\/SubtractingIntegers_49.png\" alt=\"3 red circles with a number 3 underneath\" width=\"354\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>We need to take away [latex]8[\/latex] negatives, but we have only [latex]3[\/latex].<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Add neutral pairs until there are [latex]8[\/latex] negatives to take away.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15557\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220831\/SubtractingIntegers_50.png\" alt=\"7 red circles and below there are 6 blue circles\" width=\"358\" height=\"94\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away the [latex]8[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15558\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220843\/SubtractingIntegers_51.png\" alt=\"8 red circles that are circled together. Below there are 5 blue circles\" width=\"358\" height=\"139\" \/><\/td>\n<\/tr>\n<tr>\n<td>\n<p>Find the number of counters that are left.<\/p>\n<p>There are [latex]5[\/latex] positives.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15559\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220856\/SubtractingIntegers_52.png\" alt=\"5 blue circles labeled 5 positives\" width=\"358\" height=\"76\" \/><\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]-3-\\left(-8\\right)=5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>Now that you have seen subtraction modeled with color counters,\u00a0we can move on to performing subtraction of integers without the models.<\/p>\n<section class=\"textbox example\">\n<ul id=\"fs-id2376804\">\n<li>Subtract [latex]-23 - 7[\/latex].\n<ul>\n<li>Think: We start with [latex]23[\/latex] negative counters. We have to subtract [latex]7[\/latex] positives, but there are no positives to take away. So we add [latex]7[\/latex] neutral pairs to get the [latex]7[\/latex] positives. Now we take away the [latex]7[\/latex] positives. So what\u2019s left? We have the original [latex]23[\/latex] negatives plus [latex]7[\/latex] more negatives from the neutral pair. The result is [latex]30[\/latex] negatives.\n<div style=\"text-align: center;\">[latex]-23 - 7=-30[\/latex]<\/p>\n<\/div>\n<p>Notice, that to subtract [latex]\\text{7,}[\/latex] we added [latex]7[\/latex] negatives.<\/li>\n<\/ul>\n<\/li>\n<li>Subtract [latex]30-\\left(-12\\right)[\/latex].\n<ul>\n<li>Think: We start with [latex]30[\/latex] positives. We have to subtract [latex]12[\/latex] negatives, but there are no negatives to take away. So we add [latex]12[\/latex] neutral pairs to the [latex]30[\/latex] positives. Now we take away the [latex]12[\/latex] negatives. What\u2019s left? We have the original [latex]30[\/latex] positives plus [latex]12[\/latex] more positives from the neutral pairs. The result is [latex]42[\/latex] positives.\n<div style=\"text-align: center;\">[latex]30-\\left(-12\\right)=42[\/latex]<\/p>\n<\/div>\n<p>Notice that to subtract [latex]-12[\/latex], we added [latex]12[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<p>While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.<\/p>\n<section class=\"textbox proTip\">Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows: Subtracting a number is the same as adding its opposite.<\/p>\n<div style=\"text-align: center;\">[latex]a-b=a+(-b)[\/latex]<\/div>\n<\/section>\n<p>Now let\u2019s increase the complexity of the examples a little bit. We will use the order of operations to simplify terms in parentheses before we subtract from left to right.<\/p>\n<section class=\"textbox example\">Simplify: [latex]7-\\left(-4 - 3\\right)-9[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q769817\">Show Solution<\/button><\/p>\n<div id=\"q769817\" class=\"hidden-answer\" style=\"display: none\">\nWe use the order of operations to simplify this expression, performing operations inside the parentheses first. Then we subtract from left to right.<\/p>\n<table id=\"eip-id1168469609684\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>Simplify inside the parentheses first.<\/td>\n<td>[latex]7-\\left(-4 - 3\\right)-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract from left to right.<\/td>\n<td>[latex]7-\\left(-7\\right)-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]14-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8904\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8904&theme=lumen&iframe_resize_id=ohm8904&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Applications With Subtracting Integers<\/h2>\n<p>It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it. So when we solve an application problem, we first need to determine what we are asked to find. Then we can write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.<\/p>\n<p>Geography provides an application of negative numbers with the elevations of places below sea level.<\/p>\n<section class=\"textbox example\">Dinesh hiked from Mt. Whitney, the highest point in California, to Death Valley, the lowest point. The elevation of Mt. Whitney is [latex]14,497[\/latex] feet above sea level and the elevation of Death Valley is [latex]282[\/latex] feet below sea level. What is the difference in elevation between Mt. Whitney and Death Valley?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q128798\">Show Solution<\/button><\/p>\n<div id=\"q128798\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>Step 1. What are we asked to find?<\/td>\n<td>The difference in elevation between Mt. Whitney and Death Valley<\/td>\n<\/tr>\n<tr>\n<td>Step 2. Write a phrase.<\/td>\n<td>elevation of Mt. Whitney\u2212elevation of Death Valley<\/td>\n<\/tr>\n<tr>\n<td>Step 3. Translate.<\/td>\n<td>[latex]14,497-\\left(-282\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 4. Simplify.<\/td>\n<td>[latex]14,779[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. Write a complete sentence that answers the question.<\/td>\n<td>The difference in elevation is [latex]14,779[\/latex] feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8905\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8905&theme=lumen&iframe_resize_id=ohm8905&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Managing your money can involve both positive and negative numbers.<\/p>\n<section class=\"textbox example\">Leslie has [latex]$25[\/latex] in her checking account and she has a transaction for [latex]$8[\/latex].<\/p>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li>What is the balance after the transaction?<\/li>\n<li>She has a second transaction for [latex]$20[\/latex]. What is the new balance after this transaction?<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q83739\">Show Solution<\/button><\/p>\n<div id=\"q83739\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td>What are we asked to find?<\/td>\n<td>The balance of the account<\/td>\n<\/tr>\n<tr>\n<td>Write a phrase.<\/td>\n<td>[latex]$25[\/latex] minus [latex]$8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Translate<\/td>\n<td>[latex]$25-$8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]$17[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write a sentence answer.<\/td>\n<td>The balance is [latex]$17[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>What are we asked to find?<\/td>\n<td>The new balance<\/td>\n<\/tr>\n<tr>\n<td>Write a phrase.<\/td>\n<td>[latex]$17[\/latex] minus [latex]$20[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Translate<\/td>\n<td>[latex]$17-$20[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-$3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write a sentence answer.<\/td>\n<td>She is overdrawn by [latex]$3[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8906\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8906&theme=lumen&iframe_resize_id=ohm8906&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"Lynn Marecek & MaryAnne Anthony-Smith\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/prealgebra\/pages\/3-3-subtract-integers\",\"project\":\"3.3 Subtract Integers\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/3-introduction-to-integers\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":290,"module-header":"learn_it","content_attributions":[{"type":"original","description":"Revision and Adaptation","author":"","organization":"Lumen Learning","url":"","project":"","license":"cc-by","license_terms":""},{"type":"cc-attribution","description":"Prealgebra","author":"Lynn Marecek & MaryAnne Anthony-Smith","organization":"OpenStax","url":"https:\/\/openstax.org\/books\/prealgebra\/pages\/3-3-subtract-integers","project":"3.3 Subtract Integers","license":"cc-by","license_terms":"Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/3-introduction-to-integers"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5445"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":40,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5445\/revisions"}],"predecessor-version":[{"id":15175,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5445\/revisions\/15175"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/290"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5445\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=5445"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=5445"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=5445"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=5445"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}