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### High School Mathematics10.2 Solve Absolute Value Inequalities

 The equation |x| = 5 means that the distance between x and 0 is 5. The solutions of the equation are 5 and -5 because they are the only numbers whose distance from 0 is 5. The inequality |x| < 3 means that the distance between x and 0 is less than 3, and The inequality |x| > 3 means that the distance between x and 0 is greater than 3. Solving Absolute Value Inequality: The inequality |ax+b| < c where c > 0 is equivalent to the compound inequality -c < ax+b < c The inequality |ax+b| > c where c > 0 is equivalent to the compound inequality ax+b < -c or ax + b > c In the equalities above, < can be replaced by £ and > can be replaced by ³. Examples: Solve |x| ³ 8 The solutions are x ³ 8 and x £ -8 Solve |x| £ 0.7 |x| £ 0.7 -------- original equation Rewrite as two equations x £ 0.7 or x ³ -0.7 -0.7£ x £ 0.7 Solve |-4x-5|+3 < 9 |-4x-5|+3 < 9 subtracting 3 both sides of the inequation |-4x-5|+3 -3 < 9-3 |-4x-5| < 6 -6 < -4x-5 < 6 adding 5 to the inequations -6+5 < -4x-5+5 < 6+5 -1 < -4x < 11 Dividing by -4 reverses the inequality sign 0.25 > x > -2.75 This can also be written as -2.75 < x < 0.25 Solve |10 - x| > 12 10 - x > 12 or 10 - x < -12 10 -x > 12 subtracting 10 both sides 10 -x-10 > 12-10 -x > 2 Multiplying both sides by (-1) changes the sign x < -2 10 - x < -12 subtracting 10 both sides 10 -x-10 < -12-10 -x < -22 Multiplying both sides by (-1) changes the sign x > 22 Therefore x < -2 or x > 22 |x + 2| < -1 Since |x + 2| cannot be negative, |x + 2| cannot be less than -1. So, the solution set is the empty set. Solution = { } |2y -1| ³ -4 Since |2y - 1| is always greater than or equal to 0, the solution set is {y|y is a real number} The graph is the entire number line. Directions: Solve the following questions. Also write at least 5 examples of your own. Solve them and graph the solution. Click here for additional questions less than or equal to £ greater than or equal to ³
 Q 1: Solve the inequality |-2x + 1| > 2 {x: x < -3/2 or x > 1/2}{x: x < -1/2 or x > 3/2}{x: x > -1/2 or x > 3/2}{x: x < -1 or x > 1/2} Q 2: Solve the inequality |(x - 3)/2| + 2 < 6 {x: 5 < x < -11}{x: -5 > x > 11}{x: -5 < x > 11}{x: -5 < x < 11} Q 3: |-4x-5| + 3 < 9-2.75 < x < 0.250 < x < 2-2 < x < 5-2.5 < x < 2.5 Q 4: |a| < 8-8 > a < 8-8 < a > 8-8 > a > 8-8 < a < 8 Q 5: Solve -3|2-5/4u| £ -18. Then u £ ________ or u ³ _________.-3 1/5 , 6 2/52 1/2 , 5 1/31/2, 1/3 Q 6: The inequality |3x + 6| ³ 12, the solution set is {x: x£____ or x³__}6 or -26 or 2-6 or 2-6 or -2 Q 7: |h| > 3.5all of the other -3.5 > h > 3.5h > 3.5 or h < -3.5h < -3.5 or h > 3.5 Q 8: Solve the absolute value inequality: |x + 4| > -3x > -7 or x < -1All values workx < -1 or x > -7All of the others Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!

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