4.5 Solving Absolute Value Inequalities 11/16/12

Solve |x| < 3 What value of x would make what’s inside | | 7 or -7 What numbers are less than 3 units away from 0? Answer: Numbers to the right of -3, like -2, -1 and anything to the left of 3 like 2, 1 Mathematically written: -3 < x < 3 x > -3 x < 3

Solve |x| ≤ 3 Answer: Numbers to the right of -3 including 3 and anything to the left of 3 including 3. Mathematically written: -3 ≤ x ≤ 3 x ≥ -3 x ≤ 3

Solve |x| > 4 What value of x would make what’s inside | | 7 or -7 What numbers are more than 4 units away from 0? Answer: Numbers to the right of 4, like 5, 6, 7 or anything to the left of - 4 like -5, -6, -7 Mathematically written: x 4 x > 4 x < -4

Solve |x| ≥ 4 Answer: Anything to the right of 4 including 4 or anything to the left of - 4 including -4 Mathematically written: x ≤ -4 or x ≥ 4 x ≥ 4 x ≤ -4

Summary: InequalityEquivalent FormGraph |x| < 3-3 < x < 3Shade between -3 and 3 |x| ≤ 3-3 ≤ x ≤ 3 |x| > 4x > 4 or x < -4Shade opposite direction (away from each other) from 4 and -4 |x| ≥ 4x ≥ 4 or x ≤ -4 < and ≤ are both AND problems > and ≥ are both OR problems (think greater pronounced as greatOR) “AND” “OR”

Solving more complex Inequality

Example 1 Solve an Inequality of the Form Solve. Then graph the solution. + xb ≤ c + x4 ≤ 10 – + x4 ≤ Write equivalent compound inequality. 10 ≤ – x ≤ 6 Subtract 4 from each expression. 14 ≤ 16 –. 14 – 12 – 10 – 8 – 6 – 4 – 2 – SOLUTION

Checkpoint Solve the inequality. Then graph your solution. Solve an Absolute Value Inequality 1. + x1 ≤ 4 ANSWER – x ≤ 35 ≤ 6 – 4 – 2 – < x < 9 ANSWER

Example 2 Solve an Inequality of the Form Solve. Then graph the solution. + axbc < + 2x2x37 < – Write equivalent compound inequality. + 2x2x37 < 7 < – Subtract 3 from each expression. 10 2x2x4 << – Divide each expression by 2. 5 x2 << 6 – 5 – 4 – 3 – 0122 – 1 – 3 SOLUTION

Checkpoint Solve the inequality. Then graph your solution. Solve an Absolute Value Inequality 4. 2x2x19 < – 5. 14x4x3 < + ANSWER 1 – 0.. – x0.51 << 1 ANSWER – x54 << 4 – 2 –

Example 3 Solve an Inequality of the Form Solve. Then graph the solution. + axbc≥ x –≥ FIRST INEQUALITYSECOND INEQUALITY x – ≤ – x ≥– Add 1 to each side. x2 2 1 ≤ – 4x 2 1 ≥

Checkpoint Solve the inequality. Then graph your solution. Solve an Absolute Value Inequality 7. 2x2x37 +> 8. 4x4x15 + ≥ ANSWER – 1.5 or xx 1 ≥ 1 – 012 –.. ≤ ANSWER < – 5 or x 2 > x 4 – 2 – – 5 –

Homework: 4.5 p.201 #11-20 all even even