Absolute Value Equations and Inequalities Absolute Value of

Absolute Value Equations and Inequalities

Absolute Value (of x) u u Symbol lxl The distance x is from 0 on the number line. Always positive Ex: l-3 l = 3 -4 -3 -2 -1 0 1 2

Ex: x = 5 u What are the possible values of x? x=5 or x = -5

To solve an absolute value equation: ax+b = c, where c > 0 To solve, set up 2 new equations, then solve each equation. ax + b = c or ax + b = -c ** make sure the absolute value is by itself before you split to solve.

Ex: Solve 6 x - 3 = 15 6 x-3 = 15 or 6 x = 18 or x = 3 or 6 x-3 = -15 6 x = -12 x = -2 * Plug in answers to check your solutions!

Ex: Solve 2 x + 7 - 3 = 8 Get the abs. value part by itself first! 2 x+7 = 11 Now split into 2 parts. 2 x+7 = 11 or 2 x+7 = -11 2 x = 4 or 2 x = -18 x = 2 or x = -9 Check the solutions.

Solving Absolute Value Inequalities 1. ax+b < c, where c > 0 Becomes an “and” problem Changes to: ax+b < c and ax+b > -c 2. ax+b > c, where c > 0 Becomes an “or” problem Changes to: ax+b > c or ax+b < -c “less th. AND” “great. OR”

Ex: Solve & graph. u Becomes an “and” problem -3 7 8

Solve & graph. u Get absolute value by itself first. u Becomes an “or” problem -2 3 4

Solving an Absolute Value Equation Solve x=7 or x=− 2

Solving with less than Solve

Solving with greater than Solve

Example 1: ● |2 x + 1| > 7 ● 2 x + 1 > 7 or 2 x + 1 >7 ● 2 x + 1 >7 or 2 x + 1 <-7 ● x > 3 or This is an ‘or’ statement. (Greator). Rewrite. In the 2 nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. x < -4 Graph the solution. -4 3

Example 2: This is an ‘and’ statement. (Less thand). ● |x -5|< 3 ● x -5< 3 and x -5> -3 ● ● Rewrite. In the 2 nd inequality, reverse the inequality sign and negate the right side value. x < 8 and x > 2 2<x<8 Solve each inequality. Graph the solution. 2 8

Solve the equation. |– 3 + k| = 10 This can be read as “the distance from k to – 3 is 10. ” – 3 + k = 10 or – 3 + k = – 10 Rewrite the absolute value as a disjunction. k = 13 or k = – 7 Add 3 to both sides of each equation.

Solve the equation. Isolate the absolute-value expression. Rewrite the absolute value as a disjunction. x = 16 or x = – 16 Multiply both sides of each equation by 4.

Solve the inequality. Then graph the solution. |– 4 q + 2| ≥ 10 – 4 q + 2 ≥ 10 or – 4 q + 2 ≤ – 10 Rewrite the absolute value as a disjunction. – 4 q ≥ 8 or – 4 q ≤ – 12 Subtract 2 from both sides of each inequality. q ≤ – 2 or q ≥ 3 Divide both sides of each inequality by – 4 and reverse the inequality symbols.

Solve the inequality. Then graph the solution. |3 x| + 36 > 12 Isolate the absolute value as a disjunction. |3 x| > – 24 Rewrite the absolute value as a disjunction. 3 x > – 24 or 3 x < 24 Divide both sides of each inequality by 3. x > – 8 or x < 8 The solution is all real numbers, R. (–∞, ∞) – 3 – 2 – 1 0 1 2 3 4 5 6

Solve the compound inequality. Then graph the solution set. |2 x +7| ≤ 3 2 x + 7 ≤ 3 and 2 x + 7 ≥ – 3 2 x ≤ – 4 and 2 x ≥ – 10 x ≤ – 2 and x ≥ – 5 Multiply both sides by 3. Rewrite the absolute value as a conjunction. Subtract 7 from both sides of each inequality. Divide both sides of each inequality by 2.

Solve the compound inequality. Then graph the solution set. |p – 2| ≤ – 6 and p – 2 ≥ 6 p ≤ – 4 and p ≥ 8 Multiply both sides by – 2, and reverse the inequality symbol. Rewrite the absolute value as a conjunction. Add 2 to both sides of each inequality. Because no real number satisfies both p ≤ – 4 and p ≥ 8, there is no solution. The solution set is ø.