Solving and Graphing Compound Inequalities

h ≥ 52 and h ≤ 72 Inequalities Containing and To ride a roller coaster, you must be at least 52 inches tall, and your height cannot exceed 72 inches. If h represents the height of the rider, we can write two inequalities to represent this. At least 52 inches Cannot exceed 72 inches h ≥ 52 and h ≤ 72 The inequalities h ≥ 52 and h ≤ 72 can be combined and written without using “and” as 52 ≤ h ≤ 72 Graph the inequality “sandwich” Variable is isolated

You try! Inequalities Containing and Solve, graph, and write in interval notation. 1.) -2 < x – 3 < 4 2.) -5 < 3p + 7 ≤ 22 1.) 1 < x < 7 (1, 7) 2.) -4 < p ≤ 5 (-4, 5]

Inequalities Containing or SNAKES Most snakes live where the temperature ranges from 750 F to 900 F. Write an inequality to represent temperatures where snakes will not thrive. Let t = temperature t < 75 or t > 90 Graph the inequality “torpedo”

You try! Inequalities Containing or Solve, graph, and write in interval notation. 5n – 1 < -16 or -3n – 1 < 8 The product of -5 and a number is greater than 35 or at least 10. n < -3 or n > -3 (-∞, -3) U (-3, ∞) 2. -5n > 35 or -5n ≤ 10 n < -7 or n ≥ -2 (-∞, -7) U [-2, ∞)

OPEN ENDED 1: Write a compound inequality containing and for which the graph is the empty set. Sample answer: x ≤ -4 and x ≥ 1 OPEN ENDED 2: Create an example of a compound inequality containing or that has infinitely many solutions. Sample answer: x ≤ 5 or x ≥ 1

Chemistry The acidity of the water in a swimming pool is considered normal if the average of three pH readings is between 7.2 and 7.8. The first two readings for the swimming pool are 7.4 and 7.9. What possible values for the third reading p will make the average pH normal? 7.2 ≤ (7.4 + 7.9 + p)/3 ≤ 7.8 3(7.2) ≤ 3(7.4 + 7.9 + p)/3 ≤ 3(7.8) 21.6 ≤ 15.3 + p ≤ 23.4 21.6 – 15.3 ≤ 15.3 + p – 15.3 ≤ 23.4 – 15.3 6.3 ≤ p ≤ 8.1 The value for the third reading must be between 6.3 and 8.1, inclusive.

The value for the third reading must be between 6. 3 and 8 The value for the third reading must be between 6.3 and 8.1, inclusive.

GEOMETRY The Triangle Inequality Theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side. a.) Write and solve three inequalities to express the relationships among the measures of the sides of the triangle shown above. b.) What are the possible lengths for the third side of the triangle? c.) Write a compound inequality for the possible values of x. 9 x 4 a.) x + 9 > 4, x > -5 x + 4 > 9, x > 5 4 + 9 > x, x < 13 b.) 6, 8, 7, 9, 10, 11, 12 c.) 5 < x < 13