Equations and Inequalities - About

1 absolute - value Equations and Inequalities353 Solve absolute - value absolute - value Inequalities . To solve real-lifeproblemssuch as finding thewavelengths of differentcolors of fireworks inExs. 65 you should learn itGOAL2 GOAL1 What you should absolute -ValueEquations and InequalitiesSOLVINGABSOLUTE-VALUEEQUATIO NSYou can solve some absolute - value Equations using mental math. For instance,you learned in Lesson that the equation |x|= 8 has twosolutions: 8 and solve absolute - value Equations , you can use the fact that the expression insidethe absolute value symbols can be either positive or an absolute - value EquationSolve |x 2|= |x 2|= 5, the expression x 2 can be equal to 5 or to 2 IS POSITIVEx 2 IS NEGATIVE|x 2|= 5|x 2|= 5x 2 = +5x 2 = 5x= 7x= 3 The equation has two solutions: 7 and 3. CHECK Substitute both values into the original equation.

2 |7 2|=|5|= 5| 3 2|=| 5|= 5 Solving an absolute - value EquationSolve |2x 7| 5 = the absolute - value expression on one side of the 7 IS POSITIVE2x 7 IS NEGATIVE|2x 7| 5 = 4|2x 7| 5 = 4|2x 7|= 9|2x 7|= 92x 7 = +92x 7 = 92x= 162x= 2x= 8x= 1 The equation has two solutions: 8 and 1. Check these solutions in the original 2 EXAMPLE 1 GOAL1 SOLVINGABSOLUTE-VALUEINEQUALITIESR ecall that |x| is the distance between xand 0. If |x|< 8, then any numberbetween 8 and 8 is a solution of the can use these properties to solve absolute - value Inequalities and Equations . Notice that when an absolute value is less thana number, the Inequalities areconnected by and. When an absolute value is greater thana number, theinequalities are connected by an absolute - value InequalitySolve |x 4|< 4 IS POSITIVEx 4 IS NEGATIVE|x 4|< 3|x 4|< 3x 4 < +3x 4 > 3 Reverse inequality < 7x> 1 The solution is all real numbers greater than 1 andless than 7, which can bewritten as 1 < x< 3 GOAL2354 Chapter 6 Solving and Graphing Linear InequalitiesEach absolute - value equation or inequality is rewritten as two Equations ortwo Inequalities joined by andor or.

3 |ax+ b|< cax + b< candax+ b> c. |ax+ b| cax + b candax+b c. |ax + b|= cax + b = cor ax + b = c. |ax + b|> cax + b > c orax + b < c. |ax + b| cax + b cor ax + b absolute - value Equations AND Inequalities meansmeansmeansmeansInvestigating absolute - value InequalitiesUse Guess, Check, and Revise to find values of xthat satisfy each absolute - value inequality. Graph the solution set on a number line. Then use acompound inequality to describe the solution |x|< 22.|x+ 2| 13.|x 3| 2 DevelopingConceptsACTIVITYSTUDENTHELPS tudy TipWhen you rewrite anabsolute-valueinequality, reverse theinequality symbol in theinequality involving absolute - value Equations and Inequalities355 Solving an absolute - value InequalitySolve |2x+ 1| 3 6. Then graph the +1 IS POSITIVE2x+1 IS NEGATIVE|2x+ 1| 3 6|2x+ 1| 3 6|2x+ 1| 9|2x+ 1| 92x+ 1 +92x+ 1 9 Reverse inequality 82x 10x 4x 5 The solution of |2x+ 1| 3 6 is all real numbers greater than or equal to4 orless than or equal to 5, which can be written as the compound inequalityx 5 orx an absolute - value InequalityYou work in the quality control department of a manufacturing company.

4 The diameter of a drill bit must be between inch and inch. an absolute - value inequality to represent this bit has a diameter of inch. Does it meet the requirement? drepresent the diameter (in inches) of the drill compound d halfway point: from each part of the compound d d an absolute - value inequality: |d | inequality can be read as the actual diameter must differ from inch by no more than inch. | | , the bit does meet the 5 66012345 5 4 3 2 1 EXAMPLE CONTROLINSPECTORA quality control inspectorusually works in a manufac-turing company. Experiencedworkers usually advance tomore complex ONCAREERSHOMEWORK HELPV isit our Web extra 6 Solving and Graphing Linear Inequalities1.|x+ 3|= 8 is an example of a(n) ?and |x+ 3| 8 is an example ofa(n) ?. each step you should use to solve |x+ 3|= why |x 5|< 2 means that x 5 is between 2 and the sentence using the word andor the word |x 5|< 2 means x 5<2 ?

5 X 5> |x 5|> 2 means x 5>2 ?x 5< the |n|= 57.|a|= 08.|x+ 3|= 69.|x 4|= 1010.|2n 3|+ 4 = 811.|3x+ 2|+ 2 = 5 Match the inequality with the graph of its |x+ 2| 1A. 13.|x+ 2| 1B. 14.|x+ 2|> 1C. Solve the |x+ 6|< 416.|x 2|> 917.|3x+ 1| Example 5, suppose the diameter of the drill bitcould be inch, plus or minus as much as inch. Write an absolutevalue inequality to represent this requirement. SOLVINGEQUATIONSS olve the |x|= 720.|x|= 1021.|x|= 2522.|x 4|= 6 23.|x+ 5|= 1124.|x+ 8|= 925.|x 5|= 2 26.|x 1|= 427.|x+ 3|= 928.|x+ 1|= 6 29.|x |= 730.|x+ 5|= |4x 2|= 22 32.|6x 4|= 233.|3x+ 5|= 2334.|5 4x| 3 = 4 35.|2x 4| 8 = 1036.|7x+ 3|+ 2 = 3337.|x+ |= |x | 2 = 539.|x 12 |= 25 PRACTICEANDAPPLICATIONS0 5 4 3 2 110 5 4 3 2 110 5 4 3 2 11 GUIDEDPRACTICEV ocabulary Check Concept Check Skill Check STUDENTHELPE xtra Practice to help you masterskills is on p.

6 HELPE xample 1:Exs. 19 39 Example 2:Exs. 19 39continued on p. absolute - value Equations and Inequalities357 SOLVINGINEQUALITIESS olve the |x+ 3|< 841.|2x 9| 1142.|x+ 10| 2043.|x |> 344.|4x+ 2| 1 < 545.|5x 15| 4 21 SOLVING ANDGRAPHINGS olve the inequality. Then graph the |9 + x| 747.|4 x|< 548.|x+ 12|> 3649.|x 3| 1750.|x+ 5| 151.|x+ 3| 852.|10 4x| 253.|2x+ 3|> 454.|x+ 2| 5 855.|3 + x|+ 7 < 1056.|3x+ 2| 1 1057.|5x+ 1| 8 1658.|5x+ 3| 4 959.|2x+ 5| 1 < 660.|3x 9|+ 2 > your basketball team, the starting players scoringaverages are between 8 and 22 points per game. Write an absolute -valueinequality describing the scoring averages for the test scores in your class range from 60 to 100. Writean absolute - value inequality describing the range of the test car averages 28 miles per gallon in the city. Theactual mileage varies from the average by at most 4 miles per gallon.

7 Write anabsolute- value inequality that shows the range for the mileage your car cruiser weight division in boxing is centered at 183 pounds. A boxer s weight can be as much as 7 pounds more than or less than 183 pounds. Write an absolute - value inequality for this a firework star bursts, the color of the stars is determined by the chemical compounds in the firework. The wavelengths for different colors in the spectrum are shown firework star contains strontium. When it is burned, strontium emits light at wavelengths given by |w 643|< 38. What colors could the star be? firework star contains a copper compound. When it is burned, the compound emits light at wavelengths given by |w 455|< 23. Determine the color of the firework star contains barium chlorate. When it is burned, barium chlorate emits light at wavelengths given by |w |< What color is the star?

8 Firework star contains a sodium compound. When it is burned, the compound emits light at wavelengths given by |w 600|< 5. Determine the color of the , wUltravioletw< 400 Violet400 w< 424 Blue424 w< 491 Green491 w< 575 Yellow575 w< 585 Orange585 w< 647 Red647 w< 700 Infrared700 wFIREWORKS The diagram aboveshows what happens whena firework is ONAPPLICATIONSL aunch tubeStarsSTUDENTHELPHOMEWORK HELP continued from p. 356 Example 3: Exs. 40 60 Example 4:Exs. 40 60 Example 5: Exs. 61, 62358 Chapter 6 Solving and Graphing Linear |x 7|< 6. A 6 < x< 6 B 7 < x< 7 C1 < x< 13 D 1 < x< |3x+ 3|> 12. A 5 < x< 3 B3 < x< 5 Cx> 3 or x< 5 Dx> |2x 4| 3. A2 x 7 B 12 x 72 C 72 x 12 D 72 x 27 In Exercises 72 and 73, use the diagram below. Itshows how light is absorbed in an absolute - value inequality approximating the wavelengths (w) oflight that reach a depth of 30 meters in an absolute - value inequality approximating the wavelengths (w) oflight that reach a depth of 60 meters in the sum or difference of the matrices.

9 (Review )74. + 75. SLOPE-INTERCEPTFORMR ewrite the equation in slope-intercept form.(Review , for ) + 5y= + 9y= 7y= 42 EQUATIONSG raph the equation. (Review ) +y=7 WRITINGEQUATIONSW rite the point-slope form of the equation of the linethat passes through the point and has the given slope. Then rewrite theequation in slope-intercept form. (Review )82.(0, 4), m= 383.(2, 5), m= 284.( 3, 1), m= 32 is 368 miles from New York City to Pittsburgh. You makethe trip in 6 12 hours. What was your average speed? Round your answer to the nearest mile per hour. (Review )8 5 12 10 90 4 1 6 53 174 20 MIXEDREVIEWSCIENCECONNECTIONTestPreparat ion reached by light(meters)Wavelength (nanometers)015304560400500600700 Wavelength AbsorptionLight is is visible.