{"id":671,"date":"2023-03-09T17:26:22","date_gmt":"2023-03-09T17:26:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=671"},"modified":"2025-08-23T01:19:15","modified_gmt":"2025-08-23T01:19:15","slug":"integers-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/integers-learn-it-3\/","title":{"raw":"Integers: Learn It 3","rendered":"Integers: Learn It 3"},"content":{"raw":"

Absolute Value<\/h2>\r\n

We saw that numbers such as [latex]5[\/latex] and [latex]-5[\/latex] are opposites because they are the same distance from [latex]0[\/latex] on the number line. They are both five units from [latex]0[\/latex]. The distance between [latex]0[\/latex] and any number on the number line is called the absolute value<\/strong> of that number.<\/p>\r\n

Because distance is never negative, the absolute value of any number is never negative.<\/p>\r\n

The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[\/latex] is written as [latex]|5|[\/latex], and the absolute value of [latex]-5[\/latex] is written as [latex]|-5|[\/latex] as shown below.<\/p>\r\n

\r\n[caption id=\"\" align=\"aligncenter\" width=\"440\"]\"A Figure 1. A number line displaying 5 and -5[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

\r\n
\r\n

Absolute Value<\/h3>\r\n

The absolute value<\/strong> of a number is its distance from [latex]0[\/latex] on the number line.<\/p>\r\n

 <\/p>\r\n

The absolute value of a number [latex]n[\/latex] is written as [latex]|n|[\/latex].<\/p>\r\n

 <\/p>\r\n

Absolute value can be never be negative so [latex]|n|\\ge 0[\/latex] for all numbers.<\/p>\r\n<\/div>\r\n<\/section>\r\n

[ohm2_question hide_question_numbers=1]3173[\/ohm2_question]<\/section>\r\n

Comparing Absolute Values<\/h3>\r\n

Just as we did with positive and negative numbers, we can use inequality symbols to show the ordering of absolute values. It is important to know, we treat absolute value bars just like we treat parentheses in the order of operations - we simplify the expression inside first.<\/p>\r\n

Order of operations is a set of rules in mathematics that dictates the sequence in which operations should be performed to correctly solve an expression. It is commonly remembered by the acronym PEMDAS which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
\r\n
\r\nFor instance, in the expression [latex]8 + (4 * 2)^2 \u00f7 4 - 2[\/latex] using the order of operations we must:\r\n\r\n
    \r\n\t
  1. Parentheses\/Brackets: Perform the operation inside the parentheses first.\r\n\r\n
      \r\n\t
    • This gives us: [latex]8 + (8)^2 \u00f7 4 - 2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t
    • Exponents\/Orders: Next, solve for the exponent.\r\n\r\n
        \r\n\t
      • This gives us: [latex]8 + 64 \u00f7 4 - 2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t
      • Multiplication and Division: Perform multiplication and division operations from left to right.\r\n\r\n
          \r\n\t
        • This gives us: [latex]8 + 16 - 2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t
        • Addition and Subtraction: Finally, carry out addition and subtraction from left to right.\r\n\r\n
            \r\n\t
          • This gives us: [latex]24 - 2 = 22[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n

            So, the result of the expression [latex]8 + (4 * 2)^2 \u00f7 4 - 2[\/latex] following the order of operations is [latex]22[\/latex].<\/p>\r\n<\/section>\r\n

            Fill in [latex]\\text{< },\\text{ > },\\text {or }=[\/latex] for each of the following:\r\n\r\n
              \r\n\t
            1. [latex]|-5|[\/latex] \u00a0 ___<\/strong>\u00a0 [latex]-|-5|[\/latex]<\/li>\r\n\t
            2. [latex]8[\/latex]\u00a0 ___<\/strong>\u00a0 [latex]-|-8|[\/latex]<\/li>\r\n\t
            3. [latex]-9[\/latex]\u00a0 ___<\/strong>\u00a0 [latex]- |-9|[\/latex]<\/li>\r\n\t
            4. [latex]-|-7|[\/latex]\u00a0 ___<\/strong>\u00a0 [latex]- 7[\/latex]<\/li>\r\n<\/ol>\r\n

              [reveal-answer q=\"622738\"]Show Solution[\/reveal-answer]
              \r\n[hidden-answer a=\"622738\"]<\/p>\r\n

              To compare two expressions, simplify each one first. Then compare.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
              1.<\/td>\r\n <\/td>\r\n<\/tr>\r\n
               <\/td>\r\n[latex]|-5|[\/latex]\u00a0 ___\u00a0 [latex]-|-5|[\/latex]<\/td>\r\n<\/tr>\r\n
              Simplify.<\/td>\r\n[latex]5[\/latex] ___\u00a0 [latex]- 5[\/latex]<\/td>\r\n<\/tr>\r\n
              Order.<\/td>\r\n[latex]5>-5[\/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
              2.<\/td>\r\n <\/td>\r\n<\/tr>\r\n
               <\/td>\r\n[latex]8[\/latex]\u00a0 ___\u00a0 [latex]-|-8|[\/latex]<\/td>\r\n<\/tr>\r\n
              Simplify.<\/td>\r\n[latex]8[\/latex]\u00a0 ___\u00a0 [latex]- 8[\/latex]<\/td>\r\n<\/tr>\r\n
              Order.<\/td>\r\n[latex]8>-8[\/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
              3.<\/td>\r\n <\/td>\r\n<\/tr>\r\n
               <\/td>\r\n[latex]-9[\/latex]\u00a0 ___\u00a0 [latex]-|-9|[\/latex]<\/td>\r\n<\/tr>\r\n
              Simplify.<\/td>\r\n[latex]-9[\/latex]\u00a0 ___\u00a0 [latex]- 9[\/latex]<\/td>\r\n<\/tr>\r\n
              Order.<\/td>\r\n[latex]-9=-9[\/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
              4.<\/td>\r\n <\/td>\r\n<\/tr>\r\n
               <\/td>\r\n[latex]-|-7|[\/latex]\u00a0 ___\u00a0 [latex]- 7[\/latex]<\/td>\r\n<\/tr>\r\n
              Simplify.<\/td>\r\n[latex]-7[\/latex]\u00a0 ___\u00a0 [latex]- 7[\/latex]<\/td>\r\n<\/tr>\r\n
              Order.<\/td>\r\n[latex]-7=-7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

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

              Notice that the result is negative only when there is a negative sign outside the absolute value symbol.<\/p>\r\n

              [ohm2_question hide_question_numbers=1]3174[\/ohm2_question]<\/section>\r\n

              Simplify Absolute Values<\/h3>\r\n

              Absolute value bars act like grouping symbols. First, simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.<\/p>\r\n

              Simplify:\r\n\r\n
                \r\n\t
              1. [latex]|9 - 3|[\/latex]<\/li>\r\n\t
              2. [latex]|8+7|-|5+6|[\/latex]<\/li>\r\n\t
              3. [latex]24-|19 - 3\\left(6 - 2\\right)|[\/latex]<\/li>\r\n<\/ol>\r\n

                [reveal-answer q=\"47658\"]Show Solution[\/reveal-answer]
                \r\n[hidden-answer a=\"47658\"]
                \r\nFor each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.<\/p>\r\n

                1.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
                 <\/td>\r\n[latex]|9\u22123|[\/latex]<\/td>\r\n<\/tr>\r\n
                Simplify inside the absolute value sign.<\/td>\r\n[latex]|6|[\/latex]<\/td>\r\n<\/tr>\r\n
                Take the absolute value.<\/td>\r\n[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                 <\/p>\r\n

                2.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                 <\/td>\r\n[latex]|8+7|\u2212|5+6|[\/latex]<\/td>\r\n<\/tr>\r\n
                Simplify inside each absolute value sign.<\/td>\r\n[latex]|15|\u2212|11|[\/latex]<\/td>\r\n<\/tr>\r\n
                Take the absolute value of each term.<\/td>\r\n[latex]15\u221211[\/latex]<\/td>\r\n<\/tr>\r\n
                Subtract.<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                 <\/p>\r\n

                3.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                 <\/td>\r\n[latex]24-|19 - 3\\left(6 - 2\\right)|[\/latex]<\/td>\r\n<\/tr>\r\n
                Simplify in the parentheses first.<\/td>\r\n[latex]24-|19 - 3\\left(4\\right)|[\/latex]<\/td>\r\n<\/tr>\r\n
                Multiply [latex]3\\left(4\\right)[\/latex] .<\/td>\r\n[latex]24-|19 - 12|[\/latex]<\/td>\r\n<\/tr>\r\n
                Subtract inside the absolute value sign.<\/td>\r\n[latex]24-|7|[\/latex]<\/td>\r\n<\/tr>\r\n
                Take the absolute value.<\/td>\r\n[latex]24 - 7[\/latex]<\/td>\r\n<\/tr>\r\n
                Subtract.<\/td>\r\n[latex]17[\/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]3176[\/ohm2_question]<\/section>","rendered":"

                Absolute Value<\/h2>\n

                We saw that numbers such as [latex]5[\/latex] and [latex]-5[\/latex] are opposites because they are the same distance from [latex]0[\/latex] on the number line. They are both five units from [latex]0[\/latex]. The distance between [latex]0[\/latex] and any number on the number line is called the absolute value<\/strong> of that number.<\/p>\n

                Because distance is never negative, the absolute value of any number is never negative.<\/p>\n

                The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[\/latex] is written as [latex]|5|[\/latex], and the absolute value of [latex]-5[\/latex] is written as [latex]|-5|[\/latex] as shown below.<\/p>\n

                \n
                \"A
                Figure 1. A number line displaying 5 and -5<\/figcaption><\/figure>\n<\/div>\n

                 <\/p>\n

                \n
                \n

                Absolute Value<\/h3>\n

                The absolute value<\/strong> of a number is its distance from [latex]0[\/latex] on the number line.<\/p>\n

                 <\/p>\n

                The absolute value of a number [latex]n[\/latex] is written as [latex]|n|[\/latex].<\/p>\n

                 <\/p>\n

                Absolute value can be never be negative so [latex]|n|\\ge 0[\/latex] for all numbers.<\/p>\n<\/div>\n<\/section>\n