{"id":5426,"date":"2023-06-29T17:02:08","date_gmt":"2023-06-29T17:02:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=5426"},"modified":"2025-08-23T01:20:19","modified_gmt":"2025-08-23T01:20:19","slug":"integers-learn-it-4","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/integers-learn-it-4\/","title":{"raw":"Integers: Learn It 4","rendered":"Integers: Learn It 4"},"content":{"raw":"

Adding Integers<\/h2>\r\n

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.<\/p>\r\n

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult.
\r\n
\r\nUnderstanding the addition and subtraction of negative numbers can be challenging due to their abstract nature and the scarcity of everyday contexts where these operations are used. Math educators often employ various methods, including number lines, financial scenarios, and temperature changes, to illustrate these concepts with real world examples.<\/section>\r\n

We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.<\/p>\r\n

\r\n[caption id=\"attachment_15438\" align=\"aligncenter\" width=\"300\"]\"This Figure 1. The blue counter is positive and the red counter is negative[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in the figure below.<\/p>\r\n

A blue counter represents [latex]+1[\/latex]. A red counter represents [latex]-1[\/latex]. Together they add to zero.<\/p>\r\n

\r\n[caption id=\"attachment_15439\" align=\"aligncenter\" width=\"300\"]\"This Figure 2. 1 plus -1 equals 0[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

Model the following expressions:\r\n\r\n
    \r\n\t
  1. [latex]-5+\\left(-3\\right)[\/latex]<\/li>\r\n\t
  2. [latex]-2+\\left(-4\\right)[\/latex]<\/li>\r\n\t
  3. [latex]-2+\\left(-5\\right)[\/latex]<\/li>\r\n<\/ol>\r\n

    [reveal-answer q=\"78976\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"78976\"]<\/p>\r\n

      \r\n\t
    1. \r\n
        \r\n\t
      1. \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        Interpret the expression.<\/td>\r\n[latex]-5+\\left(-3\\right)[\/latex] means the sum of [latex]-5[\/latex] and [latex]-3[\/latex].<\/td>\r\n<\/tr>\r\n
        Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\r\n\"five<\/td>\r\n<\/tr>\r\n
        Model the second number. Add [latex]3[\/latex] negatives.<\/td>\r\n\"five<\/td>\r\n<\/tr>\r\n
        Count the total number of counters.<\/td>\r\n\"eight<\/td>\r\n<\/tr>\r\n
        The sum of [latex]\u22125[\/latex] and [latex]\u22123[\/latex] is [latex]\u22128[\/latex].<\/td>\r\n[latex]-5+-3=-8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t
      2. \"Six [latex]\u22126[\/latex]<\/li>\r\n\t
      3. \"Seven[latex]\u22127[\/latex]<\/span><\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n

        [\/hidden-answer]<\/p>\r\n<\/section>\r\n

        Now let\u2019s see what happens when the signs are different.<\/p>\r\n

        Model: [latex]-5+3[\/latex][reveal-answer q=\"229832\"]Show Solution[\/reveal-answer]
        \r\n[hidden-answer a=\"229832\"]\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        Interpret the expression.<\/td>\r\n[latex]-5+3[\/latex] means the sum of [latex]-5[\/latex] and [latex]3[\/latex].<\/td>\r\n<\/tr>\r\n
        Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\r\n\"five<\/td>\r\n<\/tr>\r\n
        Model the second number. Add [latex]3[\/latex] positives.<\/td>\r\n\"five<\/td>\r\n<\/tr>\r\n
        Remove any neutral pairs.<\/td>\r\n\"five<\/td>\r\n<\/tr>\r\n
        Count the result.<\/td>\r\n\"two<\/td>\r\n<\/tr>\r\n
        The sum of [latex]\u22125[\/latex] and [latex]3[\/latex] is [latex]\u22122[\/latex].<\/td>\r\n[latex]-5+3=-2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

        [\/hidden-answer]<\/p>\r\n<\/section>\r\n

        Notice that there were more negatives than positives, so the result is negative.<\/p>\r\n

        Model each addition.\r\n\r\n
          \r\n\t
        1. [latex]4 + 2[\/latex]<\/li>\r\n\t
        2. [latex]\u22123 + 6[\/latex]<\/li>\r\n\t
        3. [latex]4 + (\u22125)[\/latex]<\/li>\r\n\t
        4. [latex]-2 + (\u22123)[\/latex]<\/li>\r\n<\/ol>\r\n

          [reveal-answer q=\"480477\"]Show Solution[\/reveal-answer]
          \r\n[hidden-answer a=\"480477\"]<\/p>\r\n

            \r\n\t
          1. \r\n\r\n\r\n\r\n\r\n\r\n\r\n
             <\/td>\r\n[latex]4+2[\/latex]<\/td>\r\n<\/tr>\r\n
            Start with [latex]4[\/latex] positives.<\/td>\r\n\"four<\/td>\r\n<\/tr>\r\n
            Add two positives.<\/td>\r\n\"four<\/td>\r\n<\/tr>\r\n
            How many do you have?<\/td>\r\n[latex]4+2=6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t
          2. \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
             <\/td>\r\n[latex]-3+6[\/latex]<\/td>\r\n<\/tr>\r\n
            Start with [latex]3[\/latex] negatives.<\/td>\r\n\"three<\/td>\r\n<\/tr>\r\n
            Add [latex]6[\/latex] positives.<\/td>\r\n\"three<\/td>\r\n<\/tr>\r\n
            Remove neutral pairs.<\/td>\r\n\u00a0\"three<\/td>\r\n<\/tr>\r\n
            How many are left?<\/td>\r\n\"three<\/td>\r\n<\/tr>\r\n
            \u00a0[latex]-3+6=3[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t
          3. \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
             <\/td>\r\n[latex]4+\\left(-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
            Start with [latex]4[\/latex] positives.<\/td>\r\n\"four<\/td>\r\n<\/tr>\r\n
            Add [latex]5[\/latex] negatives.<\/td>\r\n\"four<\/td>\r\n<\/tr>\r\n
            Remove neutral pairs.<\/td>\r\n\"four<\/td>\r\n<\/tr>\r\n
            How many are left?<\/td>\r\n\"a<\/td>\r\n<\/tr>\r\n
            \u00a0[latex]4+\\left(-5\\right)=-1[\/latex]<\/td>\r\n[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t
          4. \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
             <\/td>\r\n <\/td>\r\n<\/tr>\r\n
             <\/td>\r\n[latex]-2+\\left(-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
            Start with [latex]2[\/latex] negatives.<\/td>\r\n\"two<\/td>\r\n<\/tr>\r\n
            Add [latex]3[\/latex] negatives.<\/td>\r\n\"two<\/td>\r\n<\/tr>\r\n
            How many do you have?<\/td>\r\n[latex]-2+\\left(-3\\right)=-5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n

            [\/hidden-answer]<\/p>\r\n<\/section>\r\n

            Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions with any integers.<\/p>\r\n

            For example, if you want to add [latex]37+\\left(-53\\right)[\/latex], you don\u2019t have to count out [latex]37[\/latex] blue counters and [latex]53[\/latex] red counters.<\/p>\r\n

            Picture [latex]37[\/latex] blue counters with [latex]53[\/latex] red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because [latex]53 - 37=16[\/latex], there are [latex]16[\/latex] more negative counters.<\/p>\r\n

            \r\n

            [latex]37+\\left(-53\\right)=-16[\/latex]<\/p>\r\n<\/center>\r\n

            \r\n
            \r\n

            adding positive and negative integers<\/h3>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
            Operation<\/th>\r\nDescription<\/th>\r\nResult<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
            [latex]5 + 3[\/latex]<\/td>\r\nBoth positive, sum positive<\/td>\r\nSum is positive<\/td>\r\n<\/tr>\r\n
            [latex]-5 + (-3)[\/latex]<\/td>\r\nBoth negative, sum negative<\/td>\r\nSum is negative<\/td>\r\n<\/tr>\r\n
            [latex]5 + (-3)[\/latex]<\/td>\r\nDifferent signs, more positives<\/td>\r\nSum is positive<\/td>\r\n<\/tr>\r\n
            [latex]-5 + 3[\/latex]<\/td>\r\nDifferent signs, more negatives<\/td>\r\nSum is negative<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/section>\r\n
            \r\n

            When the signs are the same, add the numbers. When the signs are different, subtract the smaller absolute value from the larger absolute value to determine the sum's sign based on which number is larger.<\/p>\r\n<\/section>\r\n

            Simplify:\r\n\r\n
              \r\n\t
            1. [latex]19+\\left(-47\\right)[\/latex]<\/li>\r\n\t
            2. [latex]-32+40[\/latex]<\/li>\r\n<\/ol>\r\n

              [reveal-answer q=\"78916\"]Show Solution[\/reveal-answer]
              \r\n[hidden-answer a=\"78916\"]<\/p>\r\n

                \r\n\t
              1. Since the signs are different, we subtract [latex]19[\/latex] from [latex]47[\/latex]. The answer will be negative because there are more negatives than positives.
                [latex]\\begin{array}{c}19+\\left(-47\\right)\\\\ -28\\end{array}[\/latex]<\/center>\r\n

                 <\/p>\r\n<\/li>\r\n\t

              2. The signs are different so we subtract [latex]32[\/latex] from [latex]40[\/latex]. The answer will be positive because there are more positives than negatives.
                [latex]\\begin{array}{c}-32+40\\\\ 8\\end{array}[\/latex]<\/center><\/li>\r\n<\/ol>\r\n

                [\/hidden-answer]<\/p>\r\n<\/section>\r\n

                Think of positive numbers as money you have and negative numbers as money you owe. This will help you determine if your answer is positive or negative.\u00a0
                \r\n
                \r\n[latex](-4) + 7[\/latex] would be owing [latex]$4[\/latex] and having [latex]$7[\/latex], once you settle up, you still have [latex]$3[\/latex]. So the answer would be positive [latex]3[\/latex].
                \r\n
                \r\nAnother example is [latex](-3) + (-5)[\/latex]. This means you owe [latex]$3[\/latex] and you owe [latex]$5[\/latex], so you owe [latex]$8[\/latex], which would be represented by [latex]-8[\/latex].\u00a0<\/section>\r\n
                [ohm2_question hide_question_numbers=1]8899[\/ohm2_question]<\/section>\r\n

                Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers when evaluating expressions. In our first example we will evaluate a simple variable expression for a negative value.<\/p>\r\n

                Evaluate [latex]x+7\\text{ when}[\/latex]:\r\n\r\n
                  \r\n\t
                1. [latex]x=-2[\/latex]<\/li>\r\n\t
                2. [latex]x=-11[\/latex]<\/li>\r\n<\/ol>\r\n

                  [reveal-answer q=\"140148\"]Show Solution[\/reveal-answer]
                  \r\n[hidden-answer a=\"140148\"]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
                  1. Evaluate [latex]x+7[\/latex] when [latex]x=-2[\/latex]<\/td>\r\n <\/td>\r\n<\/tr>\r\n
                   <\/td>\r\n <\/td>\r\n<\/tr>\r\n
                  Substitute [latex]\\color{red}{-2}[\/latex] for [latex]x[\/latex].<\/td>\r\n\r\n

                  [latex]\\color{red}{-2}\\color{black}+7[\/latex]<\/p>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n\r\n
                  Simplify.<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
                  2. Evaluate [latex]x+7[\/latex] when [latex]x=-11[\/latex]<\/td>\r\n <\/td>\r\n<\/tr>\r\n
                   <\/td>\r\n\r\n

                  [latex]x+7[\/latex]<\/p>\r\n<\/table>\r\n<\/td>\r\n<\/tr>\r\n\r\n\r\n
                  Substitute [latex]\\color{red}{-11}[\/latex] for [latex]x[\/latex].<\/td>\r\n[latex]\\color{red}{-11}\\color{black}+7[\/latex]<\/td>\r\n<\/tr>\r\n
                  Simplify.<\/td>\r\n[latex]-4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                  [\/hidden-answer]<\/p>\r\n<\/section>\r\n

                  [ohm2_question hide_question_numbers=1]8900[\/ohm2_question]<\/section>\r\n

                  In the next example, we are give two expressions,[latex]n+1[\/latex], and [latex]-n+1[\/latex]. We will evaluate both for a negative number. This practice will help you learn how to keep track of multiple negative signs in one expression.<\/p>\r\n

                  When [latex]n=-5[\/latex], evaluate:\r\n\r\n
                    \r\n\t
                  1. [latex]n+1[\/latex]<\/li>\r\n\t
                  2. [latex]-n+1[\/latex]<\/li>\r\n<\/ol>\r\n

                    [reveal-answer q=\"57262\"]Show Solution[\/reveal-answer]
                    \r\n[hidden-answer a=\"57262\"]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                    1. Evaluate [latex]n+1[\/latex] when [latex]n=-5[\/latex]<\/td>\r\n <\/td>\r\n<\/tr>\r\n
                     <\/td>\r\n[latex]n+1[\/latex]<\/td>\r\n<\/tr>\r\n
                    Substitute [latex]\\color{red}{-5}[\/latex] for n.<\/td>\r\n[latex]\\color{red}{-5}\\color{black}+1[\/latex]<\/td>\r\n<\/tr>\r\n
                    Simplify.<\/td>\r\n[latex]-4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                    2. Evaluate [latex]-n+1[\/latex] when [latex]n=-5[\/latex]<\/td>\r\n <\/td>\r\n<\/tr>\r\n
                     <\/td>\r\n[latex]-n+1[\/latex]<\/td>\r\n<\/tr>\r\n
                    Substitute [latex]\\color{red}{-5}[\/latex] for n.<\/td>\r\n[latex]-(\\color{red}{-5}\\color{black})+1[\/latex]<\/td>\r\n<\/tr>\r\n
                    Simplify.<\/td>\r\n[latex]5+1[\/latex]<\/td>\r\n<\/tr>\r\n
                    Add.<\/td>\r\n[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                    [\/hidden-answer]<\/p>\r\n<\/section>\r\n

                    [ohm2_question hide_question_numbers=1]8901[\/ohm2_question]<\/section>\r\n

                    Applications With Adding Integers<\/h3>\r\n

                    Recall that we were introduced to some situations in everyday life that use positive and negative numbers, such as temperatures, banking, and sports. For example, a debt of [latex]$5[\/latex] could be represented as [latex]-$5[\/latex]. Let\u2019s practice translating and solving a few applications.<\/p>\r\n

                    Solving applications is easy if we have a plan. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question.<\/p>\r\n

                    The temperature in Buffalo, NY, one morning started at [latex]7[\/latex] degrees below zero Fahrenheit. By noon, it had warmed up [latex]12[\/latex] degrees. What was the temperature at noon?
                    \r\n[reveal-answer q=\"787450\"]Show Solution[\/reveal-answer]
                    \r\n[hidden-answer a=\"787450\"]
                    \r\nWe are asked to find the temperature at noon.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                    Write a phrase for the temperature.<\/td>\r\nThe temperature warmed up 12 degrees from 7 degrees below zero.<\/td>\r\n<\/tr>\r\n
                    Translate to math notation.<\/td>\r\n[latex]\u22127 + 12[\/latex]<\/td>\r\n<\/tr>\r\n
                    Simplify.<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
                    Write a sentence to answer the question.<\/td>\r\nThe temperature at noon was [latex]5[\/latex] degrees Fahrenheit.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                    [\/hidden-answer]<\/p>\r\n<\/section>\r\n

                    [ohm2_question hide_question_numbers=1]8902[\/ohm2_question]<\/section>\r\n

                    In the next example, a football team gaining and losing yardage can be represented with positive and negative numbers.<\/p>\r\n

                    A football team took possession of the football on their [latex]42[\/latex]-yard line. In the next three plays, they lost [latex]\\text{6 yards,}[\/latex] gained [latex]4[\/latex] yards and then lost [latex]8[\/latex] yards. On what yard line was the ball at the end of those three plays?
                    \r\n[reveal-answer q=\"924891\"]Show Solution[\/reveal-answer]
                    \r\n[hidden-answer a=\"924891\"]
                    \r\nWe are asked to find the yard line the ball was on at the end of three plays.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                    Write a word phrase for the position of the ball.<\/td>\r\nStart at [latex]42[\/latex], then lose [latex]6[\/latex], gain [latex]4[\/latex], lose [latex]8[\/latex].<\/td>\r\n<\/tr>\r\n
                    Translate to math notation.<\/td>\r\n[latex]42-6+4\u22128[\/latex]<\/td>\r\n<\/tr>\r\n
                    Simplify.<\/td>\r\n[latex]32[\/latex]<\/td>\r\n<\/tr>\r\n
                    Write a sentence to answer the question.<\/td>\r\nAt the end of the three plays, the ball is on the [latex]32[\/latex]-yard line.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                    [\/hidden-answer]<\/p>\r\n<\/section>","rendered":"

                    Adding Integers<\/h2>\n

                    Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.<\/p>\n

                    Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult. <\/p>\n

                    Understanding the addition and subtraction of negative numbers can be challenging due to their abstract nature and the scarcity of everyday contexts where these operations are used. Math educators often employ various methods, including number lines, financial scenarios, and temperature changes, to illustrate these concepts with real world examples.<\/section>\n

                    We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.<\/p>\n

                    \n
                    \"This
                    Figure 1. The blue counter is positive and the red counter is negative<\/figcaption><\/figure>\n<\/div>\n

                     <\/p>\n

                    If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in the figure below.<\/p>\n

                    A blue counter represents [latex]+1[\/latex]. A red counter represents [latex]-1[\/latex]. Together they add to zero.<\/p>\n

                    \n
                    \"This
                    Figure 2. 1 plus -1 equals 0<\/figcaption><\/figure>\n<\/div>\n

                     <\/p>\n

                    Model the following expressions:<\/p>\n
                      \n
                    1. [latex]-5+\\left(-3\\right)[\/latex]<\/li>\n
                    2. [latex]-2+\\left(-4\\right)[\/latex]<\/li>\n
                    3. [latex]-2+\\left(-5\\right)[\/latex]<\/li>\n<\/ol>\n