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. 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.

6 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? 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.

7 |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. 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.

8 |10 4x| = 234.|22 3n| = 535.|2n 5| = 736.|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.|12 x| 1949.

9 |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. The gathering of such information is useful when designing control panels, keyboards, gloves, and so percent of the palm widths pwere within inch of inches.

10 Write an Absolute Value inequalitythat describes these values of p. Graph the percent of the palm widths pwere within inch of inches. Write an Absolute Value inequalitythat describes these values of p. Graph the OFMEASUREMENTSYour woodshop instructor requires thatyou cut several pieces of wood within 136 inch of his specifications. Let prepresent the specification and let xrepresent the length of a cut piece of an Absolute Value inequality that describes the acceptable values of x. One piece of wood is specified to be p= 9 18 inches. Describe the acceptable lengthsfor the piece of length of a standard basketball court can vary from 84 feet to 94 feet, inclusive.