2. Aim: Absolute Value Inequalities Course: Adv. Alg. & Trig. Aim: How do we solve absolute value inequalities? Do Now: Solve and graph –8 < 3y – 20 < 52

3. Aim: Absolute Value Inequalities Course: Adv. Alg. & Trig. Absolute-Value Inequalities | ax - b | < d -d < ax - b < d + b + b +b -d + b < ax < d + b Similar, but different answers -d < ax - b < d a a a -d + b < x < d + b a a If another way: or ax - b > -d ax - b < dthen

4. Aim: Absolute Value Inequalities Course: Adv. Alg. & Trig. Absolute-Value Inequalities | x | < 8 -8 < x < 8 If another way: and x > -8 x < 8then | x + 2 | > 4 If and x + 2 < -4 x + 2 > 4then x > 2x < -6 -4 > x + 2 > 4 another way: -2 -2 -2 -6 > x > -2

5. Aim: Absolute Value Inequalities Course: Adv. Alg. & Trig. Check your answers Model Problem 1 - Method 1 Solve | 2x - 1 | < 7 and graph Graph your solution set: {x | -3 ≤ x ≤ 4} 0 1 234567 -7-6-5-4-3-2 2x - 1 > -72x - 1 < 7 True! 2(-2) - 1 > -7 2(3) - 1 < 7 -6 > -7 5 < 7 Rewrite into 2 derived inequalities 2x < 82x > -6 x > -3 x < 4 2x - 1 < 72x - 1 > -7 + 1 +1 +1 +1

6. Aim: Absolute Value Inequalities Course: Adv. Alg. & Trig. Check your answers Model Problem 1 - Method 2 Solve | 2x - 1 | < 7 and graph -7 ≤ 2x - 1 ≤ 7 Graph your solution set: {x | -3 ≤ x ≤ 4} +1 +1 +1 0 1 234567 -7-6-5-4-3-2 -6 ≤ 2x ≤ 8 2 2 2 -3 ≤ x ≤ 4 -7 ≤ 2(-2) - 1 < 7 -7 ≤ 2(3) - 1 < 7 -7≤ -5 ≤ 7 -7 ≤ 5 ≤ 7 -7 ≤ -4 - 1 < 7 True!

7. Aim: Absolute Value Inequalities Course: Adv. Alg. & Trig. Model Problem 2 -Method 1 Graph your solution set: {x | x > 2 or x < -4/3} 0 1 234567 -7-6-5-4-3-2 Solve | 3x - 1 | > 5 and graph | 3x - 1 | > 5 x > 2 x < -4/3 Solve each inequality Check your answers 3(3) - 1 > 53(-1.5) - 1<-5 8 > 5-5.5 < -5 True! Rewrite into 2 derived inequalities 3x - 1 > 53x - 1 < -5or

8. Aim: Absolute Value Inequalities Course: Adv. Alg. & Trig. Model Problem 2 - Method 2 Graph your solution set: {x | x > 2 or x < -4/3} 0 1 234567 -7-6-5-4-3-2 Solve | 3x - 1 | > 5 and graph -5 > 3x - 1 > 5 +1 +1 +1 -4 > 3x > 6 3 3 3 -4/3 > x > 2

9. Aim: Absolute Value Inequalities Course: Adv. Alg. & Trig. Conjunction/Disjunction | ax - b | < d when symbol of inequality is < or < the graph will look like this and is a conjunction of inequalities - “and”. | ax - b | > d when symbol of inequality is > or > the graph will look like this and is a disjunction of inequalities - “or”.

10. Aim: Absolute Value Inequalities Course: Adv. Alg. & Trig. Model Problems 3 - 4 3. The solution set of | x - 3 | > 5 is 1) {x|x - 2} 2) {x|x > 8 or x - 2} 4. Which graph represent the solution of the inequality | 5x - 15 | < 10? 0 1 234567 -7-6-5-4-3-2 0 1 234567 -7-6-5-4-3-2 0 1 234567 -7-6-5-4-3-2 0 1 234567 -7-6-5-4-3-2 1) 2) 3) 4)

11. Aim: Absolute Value Inequalities Course: Adv. Alg. & Trig. Model Problems 5 An archer shoots an arrow into the air with an initial velocity of 128 feet per second. Because speed is the absolute value of velocity, the arrow’s speed, s, in feet per second, after t seconds is | -32t + 128 |. Find the values of t for which s is less that 48 feet per second. s = | -32t + 128 |< 48 Rewrite into 2 derived inequalities x > 2.5 x < 5.5 Solve each inequality Check your answers -32(3) + 128 < 48-32(5) + 128 > -48 -32 > -48 True!32 < 48 0 1 234567 -7-6-5-4-3-2 -32t + 128 < 48-32t + 128 > -48 or