2-5 Warm Up Problem of the Day Lesson Presentation Solving Inequalities Containing Integers Pre-Algebra Warm Up Problem of the Day Lesson Presentation

2-5 > > < = > < Pre-Algebra 2-5 Solving Inequalities Containing Integers Warm Up Compare. Use >, <, or =. > 1. –3 –4 > 2. 6 –2 3. –8 –5 < 4. –5 –5 = > 5. 3 –3 6. –8 –2 <

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 ? Wallace

Learn to solve inequalities with integers.

When you pour salt on ice, the ice begins to melt 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.

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  –17, the ice will remain frozen.

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!

Additional Example 1A: Adding and Subtracting to Solve Inequalities Solve and graph. A. k +3 > –2 –3 k +3 > –2 Subtract 3 from both sides. k > –5 –5

Solve and graph. B. r – 9  12 r – 9  12 r – 9 + 9  12 + 9 r  21 Additional Example 1B: 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

Solve and graph. u – 5  3 u – 5 + 5  3 + 5 Add 5 to both sides. Additional Example 1C: Adding and Subtracting to Solve Inequalities Continued Solve and graph. C. u – 5  3 u – 5  3 u – 5 + 5  3 + 5 Add 5 to both sides. u  8 5 8 10

Solve and graph. D. c + 6  2 c + 6  2 –6 c  –4 –7 –4 4 Additional Example 1D: Adding and Subtracting to Solve Inequalities Continued Solve and graph. D. c + 6  2 c + 6  2 –6 Subtract 6 from both sides. c  –4 –7 –4 4

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

Solve and graph. B. f – 4  8 f – 4  8 f – 4 + 4  8 + 4 r  12 12 15 Try This: Example 1B Solve and graph. B. f – 4  8 f – 4  8 f – 4 + 4  8 + 4 Add 4 to both sides. r  12 12 15 6

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

Solve and graph. D. d + 9  6 d + 9  6 –9 d  –3 –3 –7 4 Try This: Example 1D Solve and graph. D. d + 9  6 d + 9  6 –9 Subtract 9 from both sides. d  –3 –3 –7 4

5 > –1 –1 • 5 –1 • (–1) –5 1 –5 < 1 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 5 is greater than –1. –1 • 5 –1 • (–1) Multiply both sides by –1. –5 1 > or < ? –5 < 1 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

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

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

Additional Example 2: Multiplying and Dividing to Solve Inequalities Solve and graph. A. –3y  15 Divide each side by –3;  changes to . –3y  15 –3 –7 –5 4 y  –5 B. 7m < 21 7m < 21 7 Divide each side by 7. 3 –3 5 m < 3

Solve and graph. A. –8y  24 –8y  24 –8 –3 –7 4 y  –3 9f > 45 9 5 Try This: Example 2 Solve and graph. A. –8y  24 Divide each side by –8; changes > to <. –8y  24 –8 –3 –7 4 y  –3 B. 9f > 45 9f > 45 9 Divide each side by 9. 5 10 f > 5

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

Lesson Quiz: Part 2 5. 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? x  2°F