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Solving Inequalities Containing Integers

Solving Inequalities Containing Integers. 3. = –7. x. –2. Review. Solve. x = 14. 1. t + 3 = –8. t = –11. 2. –27 = 3 b. b = –9. z = 38. 4 . z – 8 = 30. Compare, Use > , <, or = . >. >. 1. –3 –4. 2. 6 –2. 3. –8 –5. 4. –5 –5. <.

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Solving Inequalities Containing Integers

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  1. Solving Inequalities Containing Integers

  2. 3. = –7 x –2 Review Solve. x = 14 1. t + 3 = –8 t = –11 2. –27 = 3b b = –9 z = 38 4. z – 8 = 30 Compare, Use >, <, or =. > > 1. –3 –4 2. 6 –2 3. –8 –5 4. –5 –5 < = > 5. 3 –3 6. –8 –2 <

  3. Learn to solve inequalities with integers.

  4. When you pour salt on ice, the ice begins to melt. If enough salt is added, the resulting saltwater will have a freezing point of –21°C, which is much less than water’s freezing point of 0°C. Adding rock salt to the ice lowers the freezing point and helps to freeze the ice cream mixture. At its freezing point, a substance begins to freeze. To stay frozen, the substance must maintain a temperature that is less than or equal to its freezing point.

  5. If you add salt to the ice that is at a temperature of –4°C, what must the temperature change be to keep the ice from melting? This problem can be expressed as the following inequality: –4 + t  –21 When you add 4 to both sides and solve, you find that if t  –21, the ice will remain frozen.

  6. Remember! The graph of an inequality shows all of the numbers that satisfy the inequality. When graphing inequalities on a number line, use solid circles ( ) for  and  and open circles ( ) for > and <.

  7. k +3 > –2 –3 –3 Example: Adding and Subtracting to Solve Inequalities Solve and graph. A. k +3 > –2 Subtract 3 from both sides. k > –5 –5 0

  8. Example: Adding and Subtracting to Solve Inequalities Continued Solve and graph. B. r –9  12 r –9  12 r – 9 +9 12 +9 Add 9 to both sides. r 21 15 21 24

  9. Example: Adding and Subtracting to Solve Inequalities Continued Solve and graph. C. u – 5  3 u – 5  3 Add 5 to both sides. u – 5 +5 3 +5 u 8 0 5 8 10

  10. –6 –6 Example: Adding and Subtracting to Solve Inequalities Continued Solve and graph. D. c + 6  2 c + 6  2 Subtract 6 from both sides. c–4 –7 –4 0 4

  11. y + 7  –1 –7 –7 –11 –8 0 Try This Solve and graph. A. y + 7  –1 Subtract 7 from both sides. y –8

  12. 6 12 15 Try This Solve and graph. B. f –4  8 f –4  8 f – 4 +4 8 +4 Add 4 to both sides. r 12

  13. –3 0 3 8 Try This Solve and graph. C. r – 2 < 1 r – 2 < 1 r – 2 +2 < 1 +2 Add 2 to both sides. r< 3

  14. –9 –9 –3 0 –7 4 Try This Solve and graph. D. d + 9  6 d + 9  6 Subtract 9 from both sides. d–3

  15. –5 < 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 5 > –1 Sometimes you must multiply or divide to isolate the variable. Multiplying or dividing both sides of an inequality by a negative number gives a surprising result. 5 > –1 5 is greater than –1. –1 • 5 –1 • (–1) Multiply both sides by –1. –51 > or < ? –5< 1 You know –5 is less than 1, so you should use <.

  16. MULTIPLYING INEQUALITIES BY NEGATIVE INTEGERS Original Inequality Multiply/Divide Words Result Multiplying or dividing by a negative number reverses the inequality symbol. 3 > 1 –6 < –2 Multiply by –2 –4  12 1  –3 Divide by –4

  17. Helpful Hint The direction of the inequality changes only if the number you are using to multiply or divide by is negative.

  18. –3y 15 –3 –3 –5 –7 0 4 7m < 21 7 7 0 5 –3 3 Example: Multiplying and Dividing to Solve Inequalities Solve and graph. A. –3y 15 Divide each side by –3;  changes to . y–5 B. 7m < 21 Divide each side by 7. m<3

  19. –8y 24 –8 –8 –3 –7 4 0 9f > 45 9 9 0 5 10 Try This Solve and graph. A. –8y 24 Divide each side by –8; changes > to <. y–3 B. 9f > 45 Divide each side by 9. f>5

  20. –2 0 2 –3 3 0 t –3 –3 3 0 0 8 4 Lesson Quiz: Part 1 Solve and graph. 1.h + 2 < 0 h–2 2. c– 5 > –2 c 3 3. < 1 t > –3 4. 7n > 28 n > 4

  21. Lesson Quiz: Part 2 A local weather forecast stated that it would be 12°F tonight and at least 10° colder the next night. Write an inequality to show how cold it will be? 5. x 2°F

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