Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Absolute Value Inequalities 1.7 Equations and Absolute Value of a Number Recall from Section P.3 that the absolute value of a number a is given by Absolute Value of a Number It represents the distance from a to the origin on the real number line Absolute Value of a Number More generally, |x – a| is the distance between x and a on the real number line. The figure illustrates the fact that the distance between 2 and 5 is 3. Absolute Value Equations E.g. 1—Solving an Absolute Value Equation Solve the equation |2x – 5| = 3 The equation |2x – 5| = 3 is equivalent to two equations 2x – 5 = 3 or 2x = 8 or x=4 or The solutions are 1 and 4. 2x – 5 = –3 2x = 2 x=1 E.g. 2—Solving an Absolute Value Equation Solve the equation 3|x – 7| + 5 = 14 First, we isolate the absolute value on one side of the equal sign. 3|x – 7| + 5 = 14 3|x – 7| = 9 |x – 7| = 3 x–7=3 or x – 7 = –3 x = 10 or x=4 The solutions are 4 and 10. Absolute Value Inequalities Properties of Absolute Value Inequalities We use these properties to solve inequalities that involve absolute value. E.g. 3—Absolute Value Inequality Solve the inequality |x – 5| < 2 The inequality |x – 5| < 2 is equivalent to –2 < x – 5 < 2 (Property 1) 3<x<7 (Add 5) Or if you want to have two equations –2 < x – 5 x–5<2 The solution set is the open interval (3, 7). E.g. 3—Absolute Value Inequality Geometrically, the solution set consists of: All numbers x whose distance from 5 is less than 2. Solution 2 E.g. 3—Absolute Value Inequality From the figure, we see that this is the interval (3, 7). E.g. 4—Solving an Absolute Value Inequality Solve the inequality |3x + 2| ≥ 4 By Property 4, the inequality |3x + 2| ≥ 4 is equivalent to: 3x + 2 ≥ 4 3x ≥ 2 x ≥ 2/3 or 3x + 2 ≤ –4 3x ≤ –6 x ≤ –2 (Subtract 2) (Divide by 3) E.g. 4—Solving an Absolute Value Inequality So, the solution set is: {x | x ≤ –2 or x ≥ 2/3} = (-∞, -2] [2/3, ∞) E.g. 5—Piston Tolerances The specifications for a car engine indicate that the pistons have diameter 3.8745 in. with a tolerance of 0.0015 in. A tolerance - a concept which means acceptable variance, in this case that the diameters can vary from the indicated specification by as much as 0.0015 in. and still be acceptable. E.g. 5—Piston Tolerances a) Find an inequality involving absolute values that describes the range of possible diameters for the pistons. b) Solve the inequality. E.g. 5—Piston Tolerances Let d represent the actual diameter of a piston. The difference between the actual diameter (d) and the specified diameter (3.8745) is less than 0.0015. So, we have |d – 3.8745| ≤ 0.0015 E.g. 5—Piston Tolerances The Example (b) inequality is equivalent to –0.0015 ≤ d – 3.8745 ≤ 0.0015 3.8730 ≤ d ≤ 3.8760 So, acceptable piston diameters may vary between 3.8730 in. and 3.8760 in. Exercise 1.7 - 84 A company manufactures industrial laminates of thickness 0.020 in with a tolerance of 0.003 What is the inequality? Let x=0.020, then |x-0.020| −0.003 ≤ x − 0.020 ≤ 0.003 Exercise Now solve for x to find out the sizes of this laminate -0.003<= x-0.020 <= 0.003 +0.020 +0.020 0.017 <= x <= 0.023