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File: Equations And Inequalities Pdf 175874 | 7 Item Download 2023-01-28 11-57-02
198 chapter 2 solving linear equations 2 7 solve absolute value inequalities learning objectives by the end of this section you will be able to solve absolute value equations solve ...

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                          198                                                                                                                        Chapter 2 Solving Linear Equations
                            2.7    Solve Absolute Value Inequalities
                          Learning Objectives
                          By the end of this section, you will be able to:
                               Solve absolute value equations
                               Solve absolute value inequalities with “less than”
                               Solve absolute value inequalities with “greater than”
                               Solve applications with absolute value
                            Be Prepared!
                               Before you get started, take this readiness quiz.
                                     1.   Evaluate: −7.
                                                          |  |
                                          If you missed this problem, review Example 1.12.
                                     2.   Fill in <, >,      or = for each of the following pairs of numbers.
                                                                                                                           (      )
                                          ⓐ −8___− −8 ⓑ 12___− −12 ⓒ −6___−6 ⓓ − −15 ___− −15
                                              |    |         |    |                  |      |     |    |                                     |     |
                                          If you missed this problem, review Example 1.12.
                                                       14−2             (        )
                                     3.   Simplify:              8−34−1 .
                                                                |                 |
                                          If you missed this problem, review Example 1.13.
                          Solve Absolute Value Equations
                          As we prepare to solve absolute value equations, we review our definition of absolute value.
                            Absolute Value
                            The absolute value of a number is its distance from zero on the number line.
                            The absolute value of a number n is written as n and n ≥ 0 for all numbers.
                                                                                             | |        | |
                            Absolute values are always greater than or equal to zero.
                          Welearnedthatbothanumberanditsoppositearethesamedistancefromzeroonthenumberline.Sincetheyhavethe
                          same distance from zero, they have the same absolute value. For example:
                                     −5 is 5 units away from 0, so −5 = 5.
                                                                              |    |
                                       5 is 5 units away from 0, so 5 = 5.
                                                                              |  |
                          Figure 2.6 illustrates this idea.
                                                                            Figure 2.6 The numbers 5 and −5 are both five
                                                                            units away from zero.
                          Fortheequation |x| = 5, wearelookingforallnumbersthatmakethisatruestatement.Wearelookingforthenumbers
                          whosedistancefromzerois5.Wejustsawthatboth5and −5 arefiveunitsfromzeroonthenumberline.Theyarethe
                          solutions to the equation.
                                                                                        If             |x| = 5
                                                                                        then            x = −5 or x = 5
                          Thesolutioncanbesimplifiedtoasinglestatementbywriting x = ±5. Thisisread,“xisequaltopositiveornegative5”.
                          We can generalize this to the following property for absolute value equations.
                          This OpenStax book is available for free at http://cnx.org/content/col12119/1.3
                                     Chapter 2 Solving Linear Equations                                                                                                                                                                                                   199
                                        Absolute Value Equations
                                        For any algebraic expression, u, and any positive real number, a,
                                                                                                                             if                    u =a
                                                                                                                                                  |   |
                                                                                                                             then                   u = −a or u = a
                                        Remember that an absolute value cannot be a negative number.
                                         EXAMPLE 2.68
                                     Solve: ⓐ x = 8 ⓑ y = −6 ⓒ z = 0
                                                        |  |                 |  |                     | |
                                             Solution
                                     ⓐ
                                                                                                                 |x| = 8
                                      Write the equivalent equations.                                              x = −8or x = 8
                                                                                                                   x = ±8
                                     ⓑ
                                                                                                                  y = −6
                                                                                                                 |   |
                                                                                                                 No solution
                                     Since an absolute value is always positive, there are no solutions to this equation.
                                     ⓒ
                                                                                                                  z = 0
                                                                                                                 | |
                                      Write the equivalent equations.                                              z = −0orz = 0
                                      Since−0 = 0,                                                                 z = 0
                                     Both equations tell us that z = 0 and so there is only one solution.
                                                TRY IT : : 2.135                        Solve: ⓐ x = 2 ⓑ y = −4 ⓒ z = 0
                                                                                                           |   |                 |  |                     | |
                                                TRY IT : : 2.136                         Solve: ⓐ x = 11 ⓑ y = −5 ⓒ z = 0
                                                                                                           |   |                   |   |                    | |
                                     Tosolveanabsolutevalueequation,wefirstisolatetheabsolutevalueexpressionusingthesameproceduresweusedto
                                     solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.
                                         EXAMPLE 2.69                      HOW TO SOLVE ABSOLUTE VALUE EQUATIONS
                                     Solve 5x−4 −3 = 8.
                                                 |             |
                                             Solution
                    200                                                                                            Chapter 2 Solving Linear Equations
                          TRY IT : : 2.137
                                                Solve: 3x − 5 − 1 = 6.
                                                       |       |
                          TRY IT : : 2.138
                                                Solve: 4x − 3 − 5 = 2.
                                                       |       |
                    The steps for solving an absolute value equation are summarized here.
                                HOW TO : : SOLVE ABSOLUTE VALUE EQUATIONS.
                                Step 1.   Isolate the absolute value expression.
                                Step 2.   Write the equivalent equations.
                                Step 3.   Solve each equation.
                                Step 4.   Check each solution.
                      EXAMPLE 2.70
                    Solve 2|x − 7| + 5 = 9.
                         Solution
                                                                                          2|x − 7| + 5 = 9
                      Isolate the absolute value expression.                                  2|x − 7| = 4
                                                                                                |x − 7| = 2
                      Write the equivalent equations.                                 x −7 = −2 or x−7 = 2
                      Solve each equation.                                                x = 5    or      x = 9
                    This OpenStax book is available for free at http://cnx.org/content/col12119/1.3
                                     Chapter 2 Solving Linear Equations                                                                                                                                                                                                   201
                                        Check:
                                                TRY IT : : 2.139                         Solve: 3 x − 4 − 4 = 8.
                                                                                                         |          |
                                                TRY IT : : 2.140                         Solve: 2 x − 5 + 3 = 9.
                                                                                                         |          |
                                     Remember, an absolute value is always positive!
                                         EXAMPLE 2.71
                                     Solve: 2x −4 +11 = 3.
                                                  |3             |
                                             Solution
                                                                                                                                2
                                                                                                                                   x −4 +11 = 3
                                                                                                                              |3             |
                                                                                                                                           2
                                      Isolate the absolute value term.                                                                        x −4 = −8
                                                                                                                                         |3             |
                                      An absolute value cannot be negative.                                                         No solution
                                                TRY IT : : 2.141
                                                                                        Solve: 3x −5 +9 = 4.
                                                                                                      |4            |
                                                TRY IT : : 2.142
                                                                                         Solve: 5x +3 +8 = 6.
                                                                                                      |6             |
                                     Some of our absolute value equations could be of the form u = v where u and v are algebraic expressions. For
                                                                                                                                                                  |  |       |  |
                                     example, |x − 3| = 2x + 1.
                                                                           |             |
                                     Howwouldwesolve them? If two algebraic expressions are equal in absolute value, then they are either equal to each
                                     other or negatives of each other. The property for absolute value equations says that for any algebraic expression, u, and
                                     a positive real number, a, if u = a, then u = −a or u = a.
                                                                                           |   |
                                     This tell us that
                                                                                                                        if                      u = v
                                                                                                                                               |  |      |   |
                                                                                                                        then                   u = −v                or          u = v
                                     This leads us to the following property for equations with two absolute values.
                                        Equations with Two Absolute Values
                                        For any algebraic expressions, u and v,
                                                                                                                             if                     u = v
                                                                                                                                                   |  |      |   |
                                                                                                                             then                   u = −v or u = v
                                     When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.
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...Chapter solving linear equations solve absolute value inequalities learning objectives by the end of this section you will be able to with less than greater applications prepared before get started take readiness quiz evaluate if missed problem review example fill in or for each following pairs numbers simplify as we prepare our definition a number is its distance from zero on line n written and all values are always equal welearnedthatbothanumberanditsoppositearethesamedistancefromzeroonthenumberline sincetheyhavethe same they have units away so figure illustrates idea both five fortheequation x wearelookingforallnumbersthatmakethisatruestatement wearelookingforthenumbers whosedistancefromzerois wejustsawthatbothand arefiveunitsfromzeroonthenumberline theyarethe solutions equation then thesolutioncanbesimplifiedtoasinglestatementbywriting thisisread xisequaltopositiveornegative can generalize property openstax book available free at http cnx org content col any algebraic expression u ...

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