 # Solve an “and” compound inequality

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Solve an “and” compound inequality
EXAMPLE 5 Solve an “and” compound inequality Solve – 4 < 6x – 10 ≤ 14. Then graph the solution. – 4 < 6x – 10 ≤ 14 Write original inequality. – < 6x – ≤ Add 10 to each expression. 6 < 6x ≤ 24 Simplify. 1 < x ≤ 4 Divide each expression by 6. ANSWER The solutions are all real numbers greater than 1 and less than or equal to 4. The graph is shown below.

Solve an “or” compound inequality
EXAMPLE 6 Solve an “or” compound inequality Solve 3x + 5 ≤ 11 or 5x – 7 ≥ 23 . Then graph the solution. SOLUTION A solution of this compound inequality is a solution of either of its parts. First Inequality Second Inequality 3x + 5 ≤ 11 Write first inequality. 5x – 7 ≥ 23 Write second inequality. 3x ≤ 6 5x ≥ 30 Subtract 5 from each side. Add 7 to each side. x ≥ 6 x ≤ 2 Divide each side by 3. Divide each side by 5.

EXAMPLE 6 Solve an “or” compound inequality ANSWER The graph is shown below. The solutions are all real numbers less than or equal to 2 or greater than or equal to 6.

EXAMPLE 7 Write and use a compound inequality Biology A monitor lizard has a temperature that ranges from 18°C to 34°C. Write the range of temperatures as a compound inequality. Then write an inequality giving the temperature range in degrees Fahrenheit.

Write and use a compound inequality
EXAMPLE 7 Write and use a compound inequality SOLUTION The range of temperatures C can be represented by the inequality 18 ≤ C ≤ 34. Let F represent the temperature in degrees Fahrenheit. 18 ≤ C ≤ 34 Write inequality. 18 ≤ ≤ 34 5 9 ( F – 32) Substitute for C. 9 5 ( F – 32) Multiply each expression by , the reciprocal of 9 5 32.4 ≤ F – 32 ≤ 61.2 64.4 ≤ F ≤ 93.2 Add 32 to each expression.

EXAMPLE 7 Write and use a compound inequality ANSWER The temperature of the monitor lizard ranges from 64.4°F to 93.2°F.

Solve the inequality. Then graph the solution.
GUIDED PRACTICE for Examples 5,6, and 7 Solve the inequality. Then graph the solution. 9. – 1 < 2x + 7 < 19 – 1 < 2x + 7 < 19 Write original inequality. – 1– 7 < 2x + 7 – 7 < 19 – 7 Subtract 7 from each expression. – 8 < 2x < 12 Simplify. – 4 < x < 6 Divide each expression by 2. ANSWER The solutions are all real numbers greater than – 4 and less than 6.

Solve the inequality. Then graph the solution.
GUIDED PRACTICE for Examples 5,6 and 7 Solve the inequality. Then graph the solution. 10. – 8 ≤ – x – 5 ≤ 6 – 8 ≤ – x – 5 ≤ 6 Write original inequality. – 8+5 ≤ – x – 5 + 5 ≤ 6 + 5 Add 5 to each expression. – 3 ≤ – x ≤ 11 – 11 ≤ x ≤ 3 Simplify. The solutions are all real numbers greater than and equal to – 11 and less than and equal to 3. ANSWER

Solve the inequality. Then graph the solution.
GUIDED PRACTICE for Examples 5,6 and 7 Solve the inequality. Then graph the solution. 11. x + 4 ≤ 9 or x – 3 ≥ 7 SOLUTION A solution of this compound inequality is a solution of either of its parts. First Inequality Second Inequality x + 4 ≤ 9 Write first inequality. x – 3 ≥ 7 Write second inequality. x ≤ 5 Subtract 4 from each side. x ≥ 10 Add 3 to each side.

GUIDED PRACTICE for Examples 5,6 and 7 ANSWER The graph is shown below. The solutions are all real numbers. less than or equal to 5 or greater than or equal to 10.

Solve the inequality. Then graph the solution.
GUIDED PRACTICE for Examples 5,6 and 7 Solve the inequality. Then graph the solution. 12. 3x – 1 < – 1 or 2x + 5 ≥ 11 SOLUTION A solution of this compound inequality is a solution of either of its parts. First Inequality Second Inequality 3x – 1< – 1 Write first inequality. 2x + 5 ≥ 11 Write second inequality. 3x ≤ 0 2x ≥ 6 Add 1 each side . Subtract 5 from each side x ≥ 3 x ≤ 0 Divide each side by 3. Divide each side by 5.

GUIDED PRACTICE for Examples 5,6 and 7 less than 0 or greater than or equal to 3. ANSWER The graph is shown below. The solutions are all real numbers.

GUIDED PRACTICE for Examples 5,6 and 7 13.
WHAT IF? In Example 7, write a compound inequality for a lizard whose temperature ranges from 15°C to 30°C. Then write an inequality giving the temperature range in degrees Fahrenheit. SOLUTION The range of temperatures C can be represented by the inequality 15 ≤ C ≤ 30. Let F represent the temperature in degrees Fahrenheit. 15 ≤ C ≤ 30 Write inequality. 15 ≤ ≤ 30 5 9 ( F – 32) Substitute for C. 9 5 ( F – 32)

The temperature of the monitor lizard ranges from 59°F to 86°F.
GUIDED PRACTICE for Examples 5,6 and 7 Multiply each expression by , the reciprocal of 9 5 27 ≤ F – 32 ≤ 54 59 ≤ F ≤ 86 Add 32 to each expression. ANSWER The temperature of the monitor lizard ranges from 59°F to 86°F.