1. 3-2 Solving Inequalities Containing Integers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

2. Warm Up – Refresh YOUR Skills! Compare, Use >, <, or =. Course 3 3-2 Solving Inequalities Containing Integers > > = < > 1. –3 –42. 6 –2 3. –8 –54. –5 –5 5. 3 –36. –8 –2 <

3. Problem of the Day Elliot led the race; Gordon was ahead of Wallace. Labonte was directly behind Gordon and directly ahead of Marlin. Who was in last place ? Course 3 3-2 Solving Inequalities Containing Integers Wallace

4. Course 3 Lesson 3-2 Solving Inequalities by Adding and Subtracting

5. Course 3 3-2 Solving Inequalities Containing Integers 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. 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. Adding rock salt to the ice lowers the freezing point and helps to freeze the ice cream mixture.

6. 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? Course 3 3-2 Solving Inequalities Containing Integers 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.

7. Course 3 3-2 Solving Inequalities Containing Integers 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 <. Remember!

8. Course 3 3-2 Solving Inequalities Containing Integers k > –5 A. k +3 > –2 Subtract 3 from both sides. –3 k +3 > –2 Solve and graph. Additional Example 1A: Adding and Subtracting to Solve Inequalities –5 0

9. Course 3 3-2 Solving Inequalities Containing Integers B. r – 9  12 r  21 Add 9 to both sides. r – 9 + 9  12 + 9 r – 9  12 21 24 15 Additional Example 1B: Adding and Subtracting to Solve Inequalities Continued Solve and graph.

10. Course 3 3-2 Solving Inequalities Containing Integers C. u – 5  3 u  8 Add 5 to both sides. u – 5 + 5  3 + 5 u – 5  3 8 10 5 0 Solve and graph. Additional Example 1C: Adding and Subtracting to Solve Inequalities Continued

11. Course 3 3-2 Solving Inequalities Containing Integers c  –4 D. c + 6  2 Subtract 6 from both sides. –6 c + 6  2 04 –7 –4 Solve and graph. Additional Example 1D: Adding and Subtracting to Solve Inequalities Continued

12. Course 3 3-2 Solving Inequalities Containing Integers y  –8 A. y + 7  –1 Subtract 7 from both sides. –7 y + 7  –1 Try This: Example 1A 0 –11 –8 Solve and graph.

13. Course 3 3-2 Solving Inequalities Containing Integers d  –3 D. d + 9  6 Subtract 9 from both sides. –9 d + 9  6 –30 –74 Try This: Example 1B Solve and graph.

14. Course 3

15. Lesson 3-3 Solving Inequalities by Multiplying and Dividing

16. –5 1 Course 3 3-3 Solving Inequalities Containing Integers 5 is greater than –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 Multiply both sides by –1. –1 5 –1 (–1) > or < ? You know –5 is less than 1, so you should use <. –5 < 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 –5 < 1 5 > –1

17. Course 3 3-3 Solving Inequalities Containing Integers The direction of the inequality changes only if you multiply or divide by a negative on your LAST step. Helpful Hint

18. Course 3 3-3 Solving Inequalities Containing Integers A. –3y  15 y  –5 Divide each side by –3;  changes to . B. 7m < 21 m < 3m < 3 Divide each side by 7. Additional Example 2: Multiplying and Dividing to Solve Inequalities Solve and graph. 0 3–3 5 –3y  15 –3–3–3–3 7m < 21 7 7 –7 –5 04

19. Course 3 3-3 Solving Inequalities Containing Integers A. –8y  24 y  –3 Divide each side by –8; changes > to <. B. 9f > 45 f > 5f > 5 Divide each side by 9. Try This: Example 2 –8y  24 –8–8–8–8 9f > 45 9 9 –3 0 –74 0510 Solve and graph.

20. Course 3 3-3 Solving Inequalities Containing Integers A. –3/8y  24 y  –64 Multiply each side by the reciprocal; sign changes > to <. B. 0.5 f > 4 f > 8 Divide each side by.5 Try This: Example 2 Y (24/1)*(-8/3) 0.5f > 4 0.5 –34 0 –64 34 0810 Solve and graph.