5.4 Solve Compound Inequalities You will solve compound inequalities. Essential Question: How do you solve compound inequalities? You will learn how to answer this question by solving the two parts of the inequality separately.

Warm-Up Exercises 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: Graph: b. All real numbers that are less than 0 or greater than or equal to 2 Inequality: Graph: –2 < x < 3 x < 0 or x  2

Warm-Up Exercises 2. All real numbers that are greater than or equal To –3 and less than 5 Inequality: GUIDED PRACTICE for Example 1 All real numbers that are less than –1 or greater than or equal to 4 1. Inequality: = –3  x < 5 x  –3 and x < 5 x < –1 or x  4 Translate the verbal phrase into an inequality. Then graph the inequality.

Warm-Up Exercises CAMERA CARS EXAMPLE 2 Write and graph a real-world compound inequality 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.

Warm-Up Exercises EXAMPLE 2 Write and graph a real-world compound inequality 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. SOLUTION

Warm-Up Exercises SOLUTION Solve 2 < x + 5 < 9. Graph your solution. EXAMPLE 3 Solve a compound inequality with and Separate the compound inequality into two inequalities. Then solve each inequality separately. Write two inequalities. Subtract 5 from each side. Simplify. The compound inequality can be written as –3 < x < 4. 2 < x + 5 x + 5 < 9 and 2 – 5 < x + 5 – 5 x + 5 – 5 < 9 – 5 and –3 < x x < 4 and

Warm-Up Exercises EXAMPLE 3 Solve a compound inequality with and ANSWER The solutions are all real numbers greater than –3 and less than 4. Graph:

Warm-Up Exercises GUIDED PRACTICE for Example 2 and 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. 3. Investing c 4.5 ANSWER

Warm-Up Exercises EXAMPLE 3 Solve a compound inequality with and Solve the inequality. Graph your solution. GUIDED PRACTICE for Example 2 and 3 –7 < x – 5 < 4 4. –2 < x < 9 ANSWER – 6 – 4 – Graph:

Warm-Up Exercises Solve the inequality. Graph your solution. 10  2y + 4  GUIDED PRACTICE for Example 2 and 3 3  y  10 ANSWER Graph: 3

Warm-Up Exercises EXAMPLE 3 Solve a compound inequality with and Solve the inequality. Graph your solution. GUIDED PRACTICE for Example 2 and 3 –7 < –z – 1 < 3 6. –4 < z < 6 ANSWER

Warm-Up Exercises Simplify. Multiply each expression by –1 and reverse both inequality symbols. Simplify. Solve a compound inequality with and EXAMPLE 4 Solve –5  –x – 3  2. Graph your solution. –5  –x – 3  2 Write original inequality. –5 + 3  –x –  Add 3 to each expression. –2  –x  5 –1(–2)  –1(–x)  –1(5) 2  x  –5

Warm-Up Exercises Solve a compound inequality with and EXAMPLE 4 –5  x  2 Rewrite in the form a  x  b. ANSWER The solutions are all real numbers greater than or equal to –5 and less than or equal to 2.

Warm-Up Exercises Write original inequality. SOLUTION Solve a compound inequality with or EXAMPLE 5 Solve 2x Graph your solution. Solve the two inequalities separately. 2x + 3 < 9 or 3x – 6 > 12 2x + 3 – 3 < 9 – 3 or 3x – > Addition or Subtraction property of inequality 2x < 6 or 3x > 18 Simplify.

Warm-Up Exercises Solve a compound inequality with or EXAMPLE 5 2x2x < or 3x3x > Division property of inequality x < 3 or x > 6 Simplify. ANSWER The solutions are all real numbers less than 3 or greater than 6. According to the graph, is 3 a solution of the compound inequality? Explain. Does a number have to be a solution of both inequalities to be a solution of the compound inequality? Explain.

Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 Solve the inequality. Graph your solution. 7. –14 < x – 8 < –1 –6 < x < 7 ANSWER

Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 8. –1  –5t + 2  4 Solve the inequality. Graph your solution. –  t  ANSWER

Warm-Up Exercises GUIDED PRACTICE for Examples 4 and h + 1< – 5 or 2h – 5 > 7 Solve the inequality. Graph your solution. h 6 ANSWER

Warm-Up Exercises GUIDED PRACTICE for Examples 4 and c + 1  –3 or 5c – 3 > 17 Solve the inequality. Graph your solution. c  –1 or c > 4 ANSWER

Warm-Up Exercises EXAMPLE 6 Solve a multi-step problem Astronomy Identify three possible temperatures (in degrees Fahrenheit) at a landing site. Solve the inequality. Then graph your solution. Write a compound inequality that describes the possible temperatures (in degrees Fahrenheit) at a landing site. The Mars Exploration Rovers Opportunity and Spirit are robots that were sent to Mars in 2003 in order to gather geological data about the planet. The temperature at the landing sites of the robots can range from  100°C to 0°C.

Warm-Up Exercises EXAMPLE 6 Solve a multi-step problem SOLUTION Write a compound inequality. Because the temperature at a landing site ranges from –100°C to 0°C, the lowest possible temperature is –100°C, and the highest possible temperature is 0°C. STEP 1 –100  C  0 Write inequality using C. (F – 32) –100   Substitute (F – 32) for C. 9 5 Let F represent the temperature in degrees Fahrenheit, and let C represent the temperature in degrees Celsius. Use the formula 5 9 C = (F – 32).

Warm-Up Exercises –180  (F – 32 )  0 EXAMPLE 6 Solve a multi-step problem STEP 2 Solve the inequality. Then graph your solution. (F – 32) –100   Write inequality from Step 1. Multiply each expression by. 5 9 –148  F  32 Add 32 to each expression.

Warm-Up Exercises EXAMPLE 6 Solve a multi-step problem STEP 3 Identify three possible temperatures. The temperature at a landing site is greater than or equal to –148°F and less than or equal to 32°F. Three possible temperatures are –115°F, 15°F, and 32°F.

Warm-Up Exercises Mars has a maximum temperature of  7°C at the equator and a minimum temperature of –133°C at the winter pole. 11. Write and solve a compound inequality that describes the possible temperatures (in degrees Fahrenheit) on Mars. GUIDED PRACTICE for Example 6 ANSWER –133 ≤ (F– 32) ≤ 27;  ≤ F ≤

Warm-Up Exercises Graph your solution. Then identify three possible temperatures (in degrees Fahrenheit) on Mars. GUIDED PRACTICE for Example 6 ANSWER Sample answer:  100°F, 0°F, 25°F

You will solve compound inequalities. Essential Question: How do you solve compound inequalities? Compound inequalities are two inequalities joined by and or joined by or. The graph of a compound inequality with and is the intersection of the graphs of the inequalities. The graph of an inequality with or is the union of the graphs. Solve the two inequalities in the compound inequality separately. On the graph of a compound inequality with and, shade all the numbers that satisfy both inequalities together. On the graph of a compound inequality with or, shade the numbers that satisfy the separate inequalities.

Warm-Up Exercises Daily Homework Quiz 1. Solve x Graph your solution. 2. Solve 3x + 2 > –7 and 4x – 1 < –5. Graph your solution. ANSWERall real numbers less than 3 or greater than 4 ANSWERall real numbers greater than –3 and less than –1

Warm-Up Exercises Daily Homework Quiz The smallest praying mantis is 0.4 inch in length. The largest is 6 inches. Write and graph a compound inequality that describes the possible lengths L of a praying mantis. 3. ANSWER 0.4 L 6 < – < –