Section 7.6 What we are Learning: To solve open sentences involving absolute value and graph the solutions

Open Sentence: Usually an equation or inequality is described as "open" in the sense that its truth value is meaningless until its variables are replaced with specific numbers, at which point the truth value can usually be determined, and then the sentences are no longer regarded as "open” Truth Value: values that make an equation or inequality true or false

Remember: The absolute value of a number is the distance the number is from zero ∣-7∣ = 7 -7

Absolute Value Equations: Solve by graphing -or- Solve by writing them as a compound sentence and solving Remember how we wrote compound inequalities as two parts?

Graphing Example: Solve ∣x - 3∣ = 5 {-2, 8} This means that the distance between x and 3 is 5 units {-2, 8} 5 units 5 units -2 3 8

Compound Sentence Example: Solve ∣x - 3∣ = 5 This means that x -3 = 5 or –(x – 3) = 5 {-2, 8} x – 3 = 5 -(x – 3) = 5 +3 +3 -x + 3 = 5 x = 8 -3 -3 -x = 2 x = -2

Absolute Value Inequalities: ∣x∣ < n; ∣x∣ ≤ n This is the “and” version of an inequality ∣x∣ > n; ∣x∣ ≥ n This is the “or” version of an inequality Write the absolute value inequality as two separate absolute value inequalities Solve each inequality by isolating the absolute value first

Example: Solve ∣3 + 2x∣ < 11, graph the solution set {-7, 8} Why? 3 + 2x < 11 -3 – 2x < 11 -3 -3 +3 +3 2x < 8 -2x < 14 2 -2 -2 x < 4 x > -7 -7 8

Example: Solve ∣5 + 2y∣ ≥ 3 and graph the solution set {-1, -4} -5 -5 +5 +5 2y ≥ -2 -2y ≥ 8 2 2 -2 -2 y ≥ -1 y ≤ -4 -4 -1 3

Let’s Work This Together: Solve and Graph

Let’s Work This Together: Express the statement in terms of an inequality involving absolute value, do not solve The cruise control of a car set at 55mph should keep the speed s within 3mph of 55mph

Let’s Work These Together: Write an open sentence involving absolute values for each graph -8 -3 3 8 -9 -5 5 9

Homework: Page 424 17 to 37 odd 41, 43