1.7 Solving Absolute Value Equations and Inequalities

1 Page 1 of 250 Chapter 1 Equations and InequalitiesSolving Absolute ValueEquations and InequalitiesSOLVINGEQUATIONS ANDINEQUALITIESThe of a number x, written|x|, is the distance the number is from 0on a number line. Notice that the Absolute Value of a number is always Absolute Value of xcan be defined algebraically as , if xis positive|x| = 0, if x=0 x, if xis negativeTo solve an Absolute Value equation of the form |x| = cwhere c> 0, use the factthat xcan have two possible values: a positive Value cor a negative Value c. Forinstance, if |x| = 5, then x= 5 or x= an Absolute Value EquationSolve |2x 5| = the Absolute Value equation as two linear Equations and then solve eachlinear equation.

2 |2x 5| = 9 Write original 5 = 9 or 2x 5 = 9 Expression can be 9 or 14 or2x= 4 Add 5 to each 7 orx= 2 Divide each side by 2. The solutions are 7 and 2. Check these by substituting each solution into theoriginal 1absolute valueGOAL1 Solve absolutevalue Equations Absolute valueequations and Inequalities to solve real-lifeproblems,such as finding acceptableweights in Example 4. To solve real-lifeproblems, such as findingrecommended weight ranges for sports equipment in Ex. you should learn itGOAL2 GOAL1 What you should Absolute Value equation |ax + b| = c, where c> 0, is equivalent to thecompound statement ax + b = cor ax + b = AN Absolute Value EQUATION 5 4 3 2 1023145 The distance between 4and 0 is 4, so | 4| distance between 4and 0 is 4, so |4| distance between0 and itself is 0, so |0| 1 of Absolute Value Equations and Inequalities51An Absolute Value inequality such as |x 2| < 4 can be solved by rewriting it as acompound inequality.

3 In this case as 4 < x 2 < an Inequality of the Form |ax +b|<cSolve |2x+ 7| < |2x+ 7| < 11 Write original inequality. 11 < 2x+ 7 < 11 Write equivalent compound inequality. 18 < 2x< 4 Subtract 7 from each expression. 9 < x< 2 Divide each expression by 2. The solutions are all real numbers greater than 9 and less than 2. Check severalsolutions in the original inequality. The graph is shown an Inequality of the Form |ax +b| cSolve |3x 2| Absolute Value inequality is equivalent to 3x 2 8 or 3x 2 FIRST INEQUALITYSOLVE SECOND INEQUALITY3x 2 8 Write 2 83x 6 Add 2 to each 10x 2 Divide each side by 130 The solutions are all real numbers less than or equal to 2 or greater than or equal to 130.

4 Check several solutions in the original inequality. The graph is shown 3 2 102 31 2103 EXAMPLE 3 5 4 3 2 102314 6 7 8 9 10 11 EXAMPLE 2 The inequality |ax+b| < c, wherec> 0,means that ax + b is between cand c. This is equivalent to c< ax + b < c. The inequality |ax+ b| > c, wherec> 0,means that ax + b is beyond cand c. This is equivalent to ax + b < c or ax + b > the first transformation, < can be replaced by . In the secondtransformation, > can be replaced by .TRANSFORMATIONS OF Absolute Value INEQUALITIESHOMEWORK HELPV isit our Web extra 1 of 252 Chapter 1 Equations and InequalitiesREALLIFEREALLIFEM anufacturingUSINGABSOLUTEVALUE INREALLIFEIn manufacturing applications, the maximum acceptable deviation of a product fromsome ideal or average measurement is called the a Model for ToleranceA cereal manufacturer has a tolerance of ounce for a box of cereal that issupposed to weigh 20 ounces.

5 Write and solve an Absolute Value inequality thatdescribes the acceptable weights for 20 ounce The weights can range between ounces and ounces, an Absolute Value ModelQUALITYCONTROLYou are a quality control inspector at a bowling pin regulation pin must weigh between 50 ounces and 58 ounces, inclusive. Write anabsolute Value inequality describing the weights you should You should reject a bowling pin if its weight wsatisfies |w 54| > 5 EXAMPLE 4 GOAL2| | Actual weight = (ounces)Ideal weight = 20(ounces)Tolerance = (ounces)| 20| algebraic model.

6 X 20 equivalent compound x 20 to each | |>Weight of pin = (ounces)Average of extreme weights = 50 +258 = 54(ounces)Tolerance = 58 54 = 4(ounces)| 54|> 4wwToleranceAverage ofextreme weightsWeightof pinLABELSVERBALMODELPROBLEMSOLVINGSTRATE GYBOWLING Bowlingpins are made frommaple wood, either solid orlaminated. They are given atough plastic coating toresist cracking. The lighterthe pin, the easier it is toknock ONAPPLICATIONSALGEBRAICMODELPage 1 of Absolute Value Equations and is the Absolute Value of a number?

7 Absolute Value of a number cannot be negative. How, then, can the absolutevalue of abe a? an example of the Absolute Value of a number. How many other numbershave this Absolute Value ? State the number or whether the given number is a solution of the |3x+ 8| = 20; 45.|11 4x| = 7; 16.|2x 9| = 11; 17.| x+ 9| = 4; 58.|6 + 3x| = 0; 29.| 5x 3| = 8; 1 Rewrite the Absolute Value inequality as a compound |x+ 8| < 511.|11 2x| 1312.|9 x| > 2113.|x+ 5| 914.|10 3x| 1715.| 14 x+ 10|< the tolerance for the 20 ounce cereal boxes inExample 4 is now ounce.

8 Write and solve an Absolute Value inequality thatdescribes the new acceptable weights of the the Absolute Value equation as two |x 8| = 1118.|5 2x| = 1319.|6n+ 1| = 21 20.|5n 4| = 1621.|2x+ 1| = 522.|2 x| = 323.|15 2x| = 824.| 12 x+ 4|= 625.| 23 x 9|= 18 CHECKING ASOLUTIOND ecide whether the given number is a solution of |4x+ 1| = 11; 327.|8 2n| = 2; 528.|6 + 12 x|= 14; 4029.| 15 x 2|= 4; 1030.|4n+ 7| = 1; 231.| 3x+ 5| = 7; 4 SOLVINGEQUATIONSS olve the |11 + 2x| = 533.|10 4x| = 234.|22 3n| = 535.|2n 5| = 736.

9 |8x+ 1| = 2337.|30 7x| = 438.| 14 x 5|= 839.| 23 x+ 2|= 1040.| 12 x 3|= 2 REWRITINGINEQUALITIESR ewrite the Absolute Value inequality as acompound |3 + 4x| 1542.|4n 12| > 1643.|3x+ 2| < 744.|2x 1| 1245.|8 3n| 1846.|11 + 4x| < 23 PRACTICEANDAPPLICATIONSGUIDEDPRACTICEV ocabulary Check Concept Check Skill Check STUDENTHELPE xtra Practice to help you masterskills is on p. HELPE xample 1:Exs. 17 40 Examples 2, 3:Exs. 41 58 Examples 4, 5:Exs. 65 76 Page 1 of 2 Solving ANDGRAPHINGS olve the inequality. Then graph your |x+ 1| < 848.

10 |12 x| 1949.|16 x| 1050.|x+ 5| > 1251.|x 8| 552.|x 16| > 2453.|14 3x| > 1854.|4x+ 10| < 2055.|8x+ 28| 3256.|20 + 12 x|> 657.|7x+ 5| < 2358.|11 + 6x| 47 SOLVINGINEQUALITIESUse the Testfeature of a graphing calculator tosolve the inequality. Most calculators use absfor Absolute Value . Forexample, you enter |x+ 1| as abs(x+ 1).59.|x+ 1| < 360.| 23 x 13 | 13 61.|2x 4| > 1062.| 12 x 1| 363.|4x 10| > 664.|1 2x| 13 PALMWIDTHSIn Exercises 65 and 66, use the following a sampling conducted by the United States Air Force, the right-hand dimensions of 4000 Air Force men were measured.