6.4 Solving 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:

All real numbers that are less than –1 or greater than or equal to 4. GUIDED PRACTICE for Example 1 All real numbers that are less than –1 or greater than or equal to 4. 1. Inequality: x < –1 or x ≥ 4 – 2 – 1 0 1 2 3 4 5 Graph: 2. All real numbers that are greater than or equal To –3 & less than 5. Inequality: x ≥ –3 or x < 5 = –3 ≤ x < 5 Graph: – 4 – 3 – 2 – 1 0 1 2 3 4 5 6

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.

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.

Solve a compound inequality with and EXAMPLE 3 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. 23 < x and x < 4 Simplify. The compound inequality can be written as – 3 < x < 4.

EXAMPLE 3 Solve a compound inequality with and ANSWER The solutions are all real numbers greater than 23 and less than 4.

GUIDED PRACTICE for Example 2 and 3 Investing 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. SOLUTION Let c represents the change in the value from the purchase price of share all possible changes is less than –$3.00 or greater than $4.50.

So the inequality is c < –3 or c > 4.5. GUIDED PRACTICE for Example 2 and 3 So the inequality is c < –3 or c > 4.5. Graph: – 4 – 3 – 2 – 1 0 1 2 3 4 5 6

Solve a compound inequality with and for Example 2 and 3 GUIDED PRACTICE Solve a compound inequality with and for Example 2 and 3 solve the inequality. Graph your solution. –7 < x – 5 < 4 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.

The solutions are all real numbers greater than –2 & less than 9. EXAMPLE 3 GUIDED PRACTICE for Example 2 and 3 ANSWER The solutions are all real numbers greater than –2 & less than 9. – 3 – 2 – 1 0 1 2 3 4 5 Graph:

Solve the inequality. Graph your solution. GUIDED PRACTICE for Example 2 and 3 Solve the inequality. Graph your solution. 10 ≤ 2y + 4 ≤ 24 5. 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 3 ≤ y and y ≤ 10 Simplify. The compound inequality can be written as 3 ≤ y ≤ 10.

Solve a compound inequality with and for Example 2 and 3 GUIDED PRACTICE Solve a compound inequality with and for Example 2 and 3 ANSWER The solutions are all real numbers greater than 3 & less than 10. Graph: -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12

Solve a compound inequality with and for Example 2 and 3 GUIDED PRACTICE Solve a compound inequality with and for Example 2 and 3 Solve the inequality. Graph your solution. –7< –z – 1 < 3 6. 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.

Solve a compound inequality with and EXAMPLE 4 Solve a compound inequality with and 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 Simplify. – 1(– 2) – 1(– x) – 1(5) > – Multiply each expression by – 1and reverse both inequality symbols. 2 x –5 > – > – Simplify.

Solve a compound inequality with and EXAMPLE 4 Solve a compound inequality with and – 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.

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

Solve a compound inequality with or EXAMPLE 5 Solve a compound inequality with or 2x 2 6 < or 3x 3 18 > Division property of inequality x < 3 or x > 6 Simplify. ANSWER The solutions are all real numbers less than 3 or greater than 6.

Solve the inequality. Graph your solution. GUIDED PRACTICE for Examples 4 and 5 Solve the inequality. Graph your solution. 7. – 14 < x – 8 < – 1 SOLUTION – 14 < x – 8 < – 1 Write original inequality. – 14 + 8 < x – 8 + 8 < – 1 + 8 Add 8 to each expression. – 6 < x < 7 Simplify.

GUIDED PRACTICE for Examples 4 and 5 ANSWER The solutions are all real numbers greater than or equal to – 6 and less than 7. –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

GUIDED PRACTICE for Examples 4 and 5 8. – 1 ≤ – 5t + 2 ≤ 4 SOLUTION Write two inequalities. – 1 – 2 ≤ – 5t + 2 – 2 and – 5t + 2 – 2 ≤ 4 – 2 Subtract 2 from each expression. – 3 ≤ – 5t and – 5t ≤ 2 Simplify. – ≤ t ≤ 2 5 3 Rewrite in the form a ≤ x ≤ b.

GUIDED PRACTICE for Examples 4 and 5 2 5 3 ANSWER The solutions are all real numbers greater than – and less than . –2 –1 0 1 2 3 5 5 5 5 5

GUIDED PRACTICE for Examples 4 and 5 9. 3h + 1< – 5 or SOLUTION 3h + 1< – 5 or 2h – 5 > 7 Write two inequalities. 3h + 1 – 1< – 5 – 1 or 2h – 5 – 5 > 7 – 5 Addition & Subtraction property inequalities. 3h < – 6 or 2h > 12 h < – 2 or h > 6 Simplify.

GUIDED PRACTICE for Examples 4 and 5 ANSWER The solutions are all real numbers less than – 2 and greater than 6 . –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

GUIDED PRACTICE for Examples 4 and 5 10. 4c + 1 ≤ – 3 or SOLUTION 4c + 1 ≤ – 3 or 5c – 3 > 17 Write original inequality. 4c + 1 – 1 ≤ – 3 – 1 or 5c – 3 + 3> 17 +3 Addition & Subtraction property inequalities. 4c ≤ – 4 or 5c > 20 c ≤ – 1 or c > 4 Simplify.

GUIDED PRACTICE for Examples 4 and 5 ANSWER The solutions are all real numbers less than – 1 and greater than 4 . –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

EXAMPLE 6 Solve a multi-step problem Astronomy 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 . • Write a compound inequality that describes the possible temperatures (in degrees Fahrenheit) at a landing site. • Solve the inequality. Then graph your solution. • Identify three possible temperatures (in degrees Fahrenheit) at a landing site.

Solve a multi-step problem EXAMPLE 6 Solve a multi-step problem SOLUTION Let F represent the temperature in degrees Fahrenheit, and let C represent the temperature in degrees Celsius. Use the formula C = (F – 32). 5 9 STEP 1 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 5 (F – 32) for C. 9

Solve a multi-step problem EXAMPLE 6 Solve a multi-step problem 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 –180 < (F – 32 ) < 0 –148 < F < 32 Add 32 to each expression.

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.

GUIDED PRACTICE for Example 6 Mars has a maximum temperature of 7°c at the equator and a minimum temperature of –133°c at winter people. 11. Write and solve a compound inequality that describes the possible temperatures (in degree Fahrenheit) on Mars. . Graph your solution. Then identify three possible temperatures (in degrees Fahrenheit) on Mars. . SOLUTION Let f represent the temperature in degrees Fahrenheit and let C represent temperature in degrees Celsius. Use the formula C = ( f – 32). 5 9

Solve a multi-step problem for Example 6 GUIDED PRACTICE Solve a multi-step problem for Example 6 Write a compound inequality because the temperatures at a landing site ranges from –133°c to 27°c, the lowest possible temperature is –133°c and highest possible temperature is 27°c. –133°c ≤ c 7°c Write inequalities using c. ≤ 5 9 ( f – 32) 7°c –133°c substitute for c. 5 9 ( f – 32)

Solve a multi-step problem for Example 6 GUIDED PRACTICE Solve a multi-step problem for Example 6 Solve the inequality. ≤ 5 9 ( f – 32) 7°c –133°c Write inequality. –239.4 ≤ f – 32 ≤ 48.6 multiply each expression by . 9 5 –239.4+32 ≤ f – 32 + 32 ≤ 48.6 + 32 Add 32 to each expression. –207.4 ≤ f ≤ 80.6 Simplify.

Solve a multi-step problem for Example 6 GUIDED PRACTICE Solve a multi-step problem for Example 6 -300 -200 -100 0 100 200 Graph: –207.4 80.6 The Temperature at landing site is greater than or equal to –207.40f and less than or equal to 80.60f. Three possible temperatures are –100°f, 0°f, 25°f. Identify three possible temperature. ANSWER