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6.4 Solving Compound Inequalities

6.4 Solving Compound Inequalities. a. All real numbers that are greater than – 2 and less than 3. Graph:. b. All real numbers that are less than 0 or greater than or equal to 2. Graph:. EXAMPLE 1. Write and graph compound inequalities.

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6.4 Solving Compound Inequalities

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  1. 6.4 Solving Compound Inequalities

  2. a. All real numbers that are greater than –2and less than 3. Graph: b. All real numbers that are less than 0or greater than or equal to 2. Graph: EXAMPLE 1 Write and graph compound inequalities Translate the verbal phrase into an inequality. Then graph the inequality. Inequality: – 2 < x < 3 x < 0 or x ≥ 2 Inequality:

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

  4. 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.

  5. EXAMPLE 2 Write and graph a real-world compound inequality SOLUTION Let hrepresent 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. 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 x + 5 < 9 and Write two inequalities. 2 – 5< x + 5 – 5 and x + 5 – 5< 9 – 5 Subtract 5 from each side. and x < 4 23 < x Simplify. The compound inequality can be written as – 3 < x < 4.

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

  8. 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. for Example 2 and 3 GUIDED PRACTICE Investing 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.

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

  10. 4. –7 < x – 5 < 4 EXAMPLE 3 for Example 2 and 3 GUIDED PRACTICE Solve a compound inequality with and solve the inequality. Graph your solution. SOLUTION Separate the compound inequality into two inequalities. Then solve each inequality separately. –7 < x – 5 x – 5 < 4 and Write two inequalities. –7 + 5< x –5 + 5 x – 5 + 5< 4 + 5 and Add 5 to each side. and –2 < x x < 9 Simplify. The compound inequality can be written as – 2 < x < 9.

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

  12. 5. 10 ≤ 2y + 4 ≤ 24 for Example 2 and 3 GUIDED PRACTICE Solve the inequality. Graph your solution. SOLUTION Separate the compound inequality into two inequalities. Then solve each inequality separately. 2y + 4 ≤ 24 10 ≤ 2y + 4 and 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 y ≤ 10 and The compound inequality can be written as 3 ≤ y ≤ 10.

  13. 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 EXAMPLE 3 for Example 2 and 3 GUIDED PRACTICE Solve a compound inequality with and

  14. 6. –7< –z – 1< 3 EXAMPLE 3 for Example 2 and 3 GUIDED PRACTICE Solve a compound inequality with and Solve the inequality. Graph your solution. SOLUTION Separate the compound inequality into two inequalities. Then solve each inequality separately. and –z – 1 < 3 –7 < –z – 1 Write two inequalities. –7 + 1< –z – 1 + 1 and –z– 1+1 < 3 + 1 Add 1 to each side. Simplify. 6 < z and z > – 4 The compound inequality can be written as – 4 < z < 6.

  15. – 1(– 2) – 1(–x) – 1(5) > > > > – – – – 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. Multiply each expression by –1and reverse both inequality symbols. 2 x –5 Simplify.

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

  17. EXAMPLE 5 Solve a compound inequality with or Solve 2x + 3 < 9 or3x – 6 > 12. Graph your solution. SOLUTION Solve the two inequalities separately. or 2x + 3 < 9 3x – 6 > 12 Write original inequality. or 3x – 6 + 6> 12 + 6 2x + 3 – 3< 9 – 3 Addition or Subtraction property of inequality or 2x < 6 3x > 18 Simplify.

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

  19. for Examples 4 and 5 GUIDED PRACTICE 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.

  20. –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 for Examples 4 and 5 GUIDED PRACTICE ANSWER The solutions are all real numbers greater than or equal to –6and less than 7.

  21. 2 3 5 5 – ≤t ≤ for Examples 4 and 5 GUIDED PRACTICE 8. – 1 ≤ – 5t + 2 ≤ 4 SOLUTION – 1 ≤ – 5t + 2 and – 5t + 2 ≤ 4 Write two inequalities. – 1 – 2 ≤ – 5t + 2 – 2 and – 5t + 2 – 2 ≤ 4 – 2 Subtract 2 from each expression. and –3 ≤ –5t –5t ≤ 2 Simplify. Rewrite in the form a ≤ x ≤ b.

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

  23. for Examples 4 and 5 GUIDED PRACTICE or 2h – 5 > 7 9. 3h + 1< – 5 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. or 3h < – 6 2h > 12 or h > 6 h < – 2 Simplify.

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

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

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

  27. 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 . EXAMPLE 6 Solve a multi-step problem Astronomy • 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.

  28. Let Frepresent the temperature in degrees Fahrenheit, and let Crepresent the temperature in degrees Celsius. Use the formula C=(F – 32). 5 9 5 –100 < C < 0 9 –100 < < 0 (F – 32) Substitute 5 (F – 32) for C. 9 EXAMPLE 6 Solve a multi-step problem SOLUTION STEP1 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. Write inequality using C.

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

  30. 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°Fand less than or equal to 32°F. Three possible temperatures are –115°F, 15°F, and 32°F.

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

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

  33. 7°c ( f – 32) ≤ ≤ multiply each expression by . 5 9 9 5 EXAMPLE 6 for Example 6 Solve a multi-step problem GUIDED PRACTICE Solve the inequality. –133°c Write inequality. –239.4 ≤ f – 32 ≤ 48.6 –239.4+32 ≤ f – 32 + 32 ≤ 48.6 + 32 Add 32 to each expression. –207.4 ≤ f ≤ 80.6 Simplify.

  34. 80.6 –207.4 Graph: -300 -200 -100 0 100 200 ANSWER Identify three possible temperature. The Temperature at landing site is greater than or equal to –207.40fand less than or equal to 80.60f. Three possible temperatures are –100°f, 0°f, 25°f. EXAMPLE 6 for Example 6 Solve a multi-step problem GUIDED PRACTICE

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