Slide 1 : You can solve some absolute-value equations using mental math. For instance, you learned that the equation | x |? 8 has two solutions: 8 and ?8. To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative.
Slide 2 : Solve | x ? 2 | ? 5 x ? 2 IS POSITIVE | x ? 2 | ? 5 x ? 7 x ? ?3 x ? 2 IS NEGATIVE | x ? 2 | ? 5 | 7 ? 2 | ? | 5 | ? 5 | ?3 ? 2 | ? | ?5 | ? 5 The expression x ? 2 can be equal to 5 or ?5. x ? 2 ? ?5 Solve | x ? 2 | ? 5 SOLUTION x ? 2 ? ?5 The equation has two solutions: 7 and –3.
Slide 3 : Solve | 2x ? 7 | ? 5 ? 4 2x ? 7 IS POSITIVE | 2x ? 7 | ? 5 ? 4 | 2x ? 7 | ? 9 2x ? 7 ? +9 2x ? 16 2x ? 7 IS NEGATIVE | 2x ? 7 | ? 5 ? 4 | 2x ? 7 | ? 9 2x ? 7 ? ?9 2x ? ?2 x ? ?1 Isolate the absolute value expression on one side of the equation. SOLUTION Solve | 2x ? 7 | ? 5 ? 4 x ? 8
Slide 4 : Recall that ?x ?is the distance between x and 0. If ?x ?? 8, then any number between ?8 and 8 is a solution of the inequality. You can use the following properties to solve absolute-value inequalities and equations. Recall that | x | is the distance between x and 0. If | x | ? 8, then any number between ?8 and 8 is a solution of the inequality.
Slide 5 : a x ? b ? c and a x ? b ? ? c. a x ? b ? c and a x ? b ? ? c. a x ? b ? c or a x ? b ? ? c. a x ? b ? c or a x ? b ? ? c. a x ? b ? c or a x ? b ? ? c. | a x ? b | ? c | a x ? b | ? c | a x ? b | ? c | a x ? b | ? c | a x ? b | ? c When an absolute value is less than a number, the inequalities are connected by and. When an absolute value is greater than a number, the inequalities are connected by or. SOLVING ABSOLUTE-VALUE INEQUALITIES
Slide 6 : Solve | x ? 4 | < 3 x ? 4 IS POSITIVE x ? 4 IS NEGATIVE | x ? 4 | ? 3 x ? 4 ? ?3 x ? 7 | x ? 4 | ? 3 x ? 4 ? ?3 x ? 1 Reverse inequality symbol. This can be written as 1 ? x ? 7. The solution is all real numbers greater than 1 and less than 7.
Slide 7 : 2x ? 1 ? ?9 | 2x ? 1 | ?3 ? 6 | 2x ? 1 | ? 9 2x ? ?10 2x + 1 IS NEGATIVE x ? ?5 Solve | 2x ? 1 | ?3 ? 6 and graph the solution. | 2x ? 1 | ? 3 ? 6 | 2x ? 1 | ? 9 2x ? 1 ? +9 2x ? 8 2x + 1 IS POSITIVE x ? 4 Reverse inequality symbol. The solution is all real numbers greater than or equal to 4 or less than or equal to ? 5. This can be written as the compound inequality x ? ? 5 or x ? 4. ? 5 4.
Slide 8 : You work in the quality control department of a manufacturing company. The diameter of a drill bit must be between 0.62 and 0.63 inch.
Slide 9 : The diameter of a drill bit must be between 0.62 and 0.63 inch.
a. Write an absolute-value inequality to represent this requirement. Let d represent the diameter (in inches) of the drill bit. Write a compound inequality. 0.62 ? d ? 0.63 Find the halfway point. 0.625 Subtract 0.625 from each part of the compound inequality. 0.62 ? 0.625 ? d ? 0.625 ? 0.63 ? 0.625 ?0.005 ? d ? 0.625 ? 0.005 Rewrite as an absolute-value inequality. | d ? 0.625 | ? 0.005
Slide 10 : The diameter of a drill bit must be between 0.62 and 0.63 inch.
b. Does a bit with a diameter of 0.623 meet the requirement? | d ? 0.625 | ? 0.005 | 0.623 ? 0.625 | ? 0.005 | ?0.002 | ? 0.005 0.002 ? 0.005