6.4

Warm Up

Lesson Quiz

Lesson Presentation

Solve Compound Inequalities

6.4 Warm-Up

ANSWER all real numbers less than –2

1. 8 > x + 10

ANSWER

6 ≤ 2x – 4 2.

all real numbers greater than or equal to 5

Solve the inequality.

ANSWER at most 15 days

3.

You estimate you can read at least 8 history text pages per day. What are the possible numbers of day it will take you to read at most 118 pages?

6.4 Example 1

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

6.4 Guided Practice

2. All real numbers that are greater than or equal To –3 and less than 5

Inequality:

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.

6.4 Example 2

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.

6.4 Example 2

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

6.4 Guided Practice

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 < –3 or c > 4.5 ANSWER

6.4 Example 3

SOLUTION

Solve 2 < x + 5 < 9. Graph your solution.

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

6.4 Example 3

ANSWER

The solutions are all real numbers greater than –3 and less than 4.

Graph:

6.4 Guided Practice

Solve the inequality. Graph your solution. –7 < x – 5 < 4 4.

–2 < x < 9 ANSWER

– 6 – 4 – 2 0 2 4 6 8 10

9 Graph:

10 ≤ 2y + 4 ≤ 24 5.

3 ≤ y ≤ 10 ANSWER

0 2 4 6 8 10 12 Graph:

3

–7 < –z – 1 < 3 6.

–4 < z < 6 ANSWER

6.4 Example 4

Simplify.

Multiply each expression by –1 and reverse both inequality symbols.

Simplify.

Solve –5 ≤ –x – 3 ≤ 2. Graph your solution.

–5 ≤ –x – 3 ≤ 2 Write original inequality.

–5 + 3 ≤ –x – 3 + 3 ≤ 2 + 3 Add 3 to each expression.

–2 ≤ –x ≤ 5

–1(–2) ≥ –1(–x) ≥ –1(5)

2 ≥ x ≥ –5

–5 ≤ x ≤ 2 Rewrite in the form a ≤ x ≤ b.

6.4 Example 4

ANSWER

The solutions are all real numbers greater than or equal to –5 and less than or equal to 2.

6.4 Guided Practice

Solve the inequality. Graph your solution.

7. –14 < x – 8 < –1

–6 < x < 7 ANSWER

8. –1 ≤ –5t + 2 ≤ 4

– ≤ t ≤ 2 5

3 5 ANSWER

6.4 Example 5

Write original inequality.

SOLUTION

Solve 2x + 3 < 9 or 3x – 6 > 12. Graph your solution.

Solve the two inequalities separately.

2x + 3 < 9 or 3x – 6 > 12

2x + 3 – 3 < 9 – 3 or 3x – 6 + 6 > 12 + 6 Addition or Subtraction property of inequality

2x < 6 or 3x > 18 Simplify.

6.4 Example 5

2x 2

6 2 < or 3x

3 18 3 > Division property

of inequality

x < 3 or x > 6 Simplify.

ANSWER

The solutions are all real numbers less than 3 or greater than 6.

6.4 Guided Practice

9. 3h + 1< – 5 or 2h – 5 > 7

Solve the inequality. Graph your solution.

h < –2 or h > 6 ANSWER

10. 4c + 1 ≤ –3 or 5c – 3 > 17

c ≤ –1 or c > 4 ANSWER

6.4 Example 6

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.

6.4 Example 6

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.

–100 ≤ C ≤ 0 Write inequality using C.

(F – 32) –100 ≤ ≤ 0 5 9

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).

STEP 1

6.4 Example 6

–180 ≤ (F – 32 ) ≤ 0

STEP 2

Solve the inequality. Then graph your solution.

(F – 32) –100 ≤ ≤ 0 5 9 Write inequality from Step 1.

Multiply each expression by . 5 9

–148 ≤ F ≤ 32 Add 32 to each expression.

6.4 Example 6

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.

6.4 Guided Practice

Mars has a maximum temperature of 27°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 degree Fahrenheit) on Mars.

ANSWER –133 ≤ (F– 32) ≤ 27; -207.4 ≤ F ≤ 80.6 5 9

• Graph your solution. Then identify three possible temperatures (in degrees Fahrenheit) on Mars.

ANSWER

Sample answer: -100°F, 0°F, 25°F

6.4 Lesson Quiz

1. Solve x + 4 < 7 or 2x – 5 > 3. Graph your solution.

2. Solve 3x + 2 > –7 and 4x – 1 < –5. Graph your solution.

ANSWER all real numbers less than 3 or greater than 4

ANSWER all real numbers greater than –3 and less than –1

6.4 Lesson 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 < – < –