Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. x ≤ 7 23 < –2x + 3 Solve each inequality and graph the solutions. 3. 4x + 1 ≤ 25 x ≤ 6 4. 0 ≥ 3x + 3 –1 ≥ x

Objectives Solve compound inequalities with one variable. Graph solution sets of compound inequalities with one variable.

When two simple inequalities are combined into one statement by the words AND or OR, the result is called a compound inequality.

Example 1: Chemistry Application The pH level of a popular shampoo is between 6.0 and 6.5 inclusive. Write a compound inequality to show the pH levels of this shampoo. Graph the solutions. Let p be the pH level of the shampoo. 6.0 is less than or equal to pH level 6.5 6.0 ≤ p ≤ 6.5 6.0 ≤ p ≤ 6.5 5.9 6.1 6.2 6.3 6.0 6.4 6.5

Let c be the chlorine level of the pool. Example 2 The free chlorine in a pool should be between 1.0 and 3.0 parts per million inclusive. Write a compound inequality to show the levels that are within this range. Graph the solutions. Let c be the chlorine level of the pool. 1.0 is less than or equal to chlorine 3.0 1.0 ≤ c ≤ 3.0 1.0 ≤ c ≤ 3.0 2 3 4 1 5 6

Example 3: Solving Compound Inequalities Involving AND Solve the compound inequality and graph the solutions. –5 < x + 1 < 2 Since 1 is added to x, subtract 1 from each part of the inequality. –5 < x + 1 < 2 –1 – 1 – 1 –6 < x < 1 Graph the intersection by finding where the two graphs overlap. –10 –8 –6 –4 –2 2 4 6 8 10

Example 4: Solving Compound Inequalities Involving AND Solve the compound inequality and graph the solutions. 8 < 3x – 1 ≤ 11 8 < 3x – 1 ≤ 11 +1 +1 +1 9 < 3x ≤ 12 Since 1 is subtracted from 3x, add 1 to each part of the inequality. Since x is multiplied by 3, divide each part of the inequality by 3 to undo the multiplication. 3 < x ≤ 4 –5 –4 –3 –2 –1 1 2 3 4 5

CLASSWORK Worksheet

WARM-UP   x = 4  

Example 1: Solving Compound Inequalities Involving OR Solve the inequality and graph the solutions. 8 + t ≥ 7 OR 8 + t < 2 8 + t ≥ 7 OR 8 + t < 2 Solve each simple inequality. –8 –8 –8 −8 t ≥ –1 OR t < –6 –10 –8 –6 –4 –2 2 4 6 8 10

Example 2: Solving Compound Inequalities Involving OR Solve the inequality and graph the solutions. 4x ≤ 20 OR 3x > 21 4x ≤ 20 OR 3x > 21 x ≤ 5 OR x > 7 Solve each simple inequality. –10 –8 –6 –4 –2 2 4 6 8 10

Solve the compound inequality and graph the solutions. Example 3 Solve the compound inequality and graph the solutions. 2 +r < 12 OR r + 5 > 19 2 +r < 12 OR r + 5 > 19 Solve each simple inequality. –2 –2 –5 –5 r < 10 OR r > 14 –4 –2 2 4 6 8 10 12 14 16

Solve the compound inequality and graph the solutions. Example 4 Solve the compound inequality and graph the solutions. 7x ≥ 21 OR 2x < –2 7x ≥ 21 OR 2x < –2 x ≥ 3 OR x < –1 Solve each simple inequality. –5 –4 –3 –2 –1 1 2 3 4 5

CLASSWORK Worksheet

WARM-UP   x = 5  

Example 1: Writing a Compound Inequality from a Graph Write the compound inequality shown by the graph. The compound inequality is x ≤ –8 OR x > 0.

Example 2: Writing a Compound Inequality from a Graph Write the compound inequality shown by the graph. The compound inequality is m > –2 AND m < 5 (or -2 < m < 5).

Example 3 Write the compound inequality shown by the graph. The compound inequality is –9 < x AND x < –2 (or –9 < x < –2).

Example 4 Write the compound inequality shown by the graph. The compound inequality is x ≤ –3 OR x ≥ 2.

CLASSWORK Worksheet