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6.3 Students will be able to solve compound inequalities. Warm-up Use the numbers: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 Which numbers are less than or equal to -1 and greater than or equal to -2? -2, -1 2. Which numbers are greater than 1 or less than -3? -5, -4, 2, 3, 4, 5 3. Which numbers are less than or equal to -2 -5, -4, -3, -2 and less than or equal to 2? 4. Which numbers are greater than -1 or greater than 3? 0, 1, 2, 3, 4, 5 1. 6.3 Students will be able to solve compound inequalities. Daily Homework Quiz For use after Lesson 6.2 Solve the inequality. Graph your solution. 1. – 72 < 8p ANSWER 2. p>–9 w > –5 – –6 ANSWER W< – 30 6.3 Students will be able to solve compound inequalities. EXAMPLE 1 Write and graph compound inequalities Translate the verbal phrase into an inequality. Then graph the inequality. a. All real numbers that are greater than – 2 and less than 3. Inequality: – 2 < x < 3 Graph: b. All real numbers that are less than 0 or greater than or equal to 2. Inequality: x < 0 or x ≥ 2 Graph: 6.3 Students will be able to solve compound inequalities. GUIDED PRACTICE Example 1 1. All real numbers that are less than –1 or greater than or equal to 4. Inequality: x < –1 or x ≥ 4 2. All real numbers that are greater than or equal To –3 and less than 5. Inequality: x ≥ –3 and x < 5 = –3 ≤ x < 5 6.3 Students will be able to solve compound inequalities. EXAMPLE 2 Write and graph a real-world compound inequality CAMERA CARS A crane sits on top of a camera car and faces toward the front. The crane’s maximum height and minimum height above the ground are shown. Write and graph a compound inequality that describes the possible heights of the crane. 6.3 Students will be able to solve compound inequalities. EXAMPLE 2 Write and graph a real-world compound inequality SOLUTION Let h represent the height (in feet) of the crane. All possible heights are greater than or equal to 4 feet and less than or equal to 18 feet. So, the inequality is 4 ≤ h ≤ 18. 6.3 Students will be able to solve compound inequalities. Solve a compound inequality with and Solve 2 < x + 5 < 9. Graph your solution. SOLUTION Separate the compound inequality into two inequalities. Then solve each inequality separately. 2 < x + 5 and x+5<9 Write two inequalities. 2 – 5 < x + 5 – 5 and x + 5 – 5 < 9 – 5 Subtract 5 from each side. –3 < x and x<4 Simplify. The compound inequality can be written as – 3 < x < 4. 6.3 Students will be able to solve compound inequalities. EXAMPLE 3 Solve a compound inequality with and ANSWER The solutions are all real numbers greater than –3 and less than 4. Graph: 6.3 Students will be able to solve compound inequalities. for Example 2 and 3 GUIDED PRCTICE Investing 3. An investor buys shares of a stock and will sell them if the change c in value from the purchase price of a share is less than –$3.00 or greater than $4.50. Write and graph a compound inequality that describes the changes in value for which the shares will be sold. SOLUTION Let c represent the change in the value from the purchase price of the shares where all possible changes are less than –$3.00 or greater than $4.50. 6.3 Students will be able to solve compound inequalities. for Example 2 and 3 ANSWER So the inequality is c < –3 or c > 4.5. 6.3 Students will be able to solve compound inequalities. EXAMPLE 3 Solve a compound inequality with and Solve the inequality. Graph your solution. 4. –7 < x – 5 < 4 SOLUTION Separate the compound inequality into two inequalities. Then solve each inequality separately. –7 < x – 5 and x–5<4 Write two inequalities. –7 + 5 < x –5 + 5 and x – 5 + 5 < 4 + 5 Add 5 to each side. –2 < x and x<9 Simplify. The compound inequality can be written as – 2 < x < 9. 6.3 Students will be able to solve compound inequalities. EXAMPLE 3 for Example 2 and 3 ANSWER The solutions are all real numbers greater than –2 and less than 9. 9 Graph: –6 –4 –2 0 2 4 6 8 10 6.3 Students will be able to solve compound inequalities. for Example 2 and 3 GUIDED PRACTICE Solve the inequality. Graph your solution. 5. 10 ≤ 2y + 4 ≤ 24 SOLUTION Separate the compound inequality into two inequalities. Then solve each inequality separately. 10 ≤ 2y + 4 and 2y + 4 ≤ 24 Write two inequalities. 10 – 4 ≤ 2y + 4 – 4 and 2y + 4 – 4 ≤ 24 – 4 Subtract 4 from each side. 6 ≤ 2y and 2y ≤ 20 Simplify. 3 ≤ y and y ≤ 10 The compound inequality can be written as 3 ≤ y ≤ 10. 6.3 Students will be able to solve compound inequalities. EXAMPLE 3 Solve a compound inequality with and ANSWER The solutions are all real numbers greater than or equal to 3 and less than or equal to 10. 3 Graph: 0 2 4 6 8 10 12 6.3 Students will be able to solve compound inequalities. Solve a compound inequality with and Solve the inequality. Graph your solution. 6. –7< –z – 1 < 3 SOLUTION Separate the compound inequality into two inequalities. Then solve each inequality separately. –7 < –z – 1 and –z – 1 < 3 Write two inequalities. –7 + 1< –z – 1 + 1 and –z – 1 + 1 < 3 + 1 Add 1 to each side. 6< z and z > – 4 Simplify. The compound inequality can be written as – 4 < z < 6.