Inequalities & Integers Learn to solve inequalities with 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!

Solve and graph. A. k +3 > –2 k +3 > –2 –3 k > –5 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 Add 9 to both sides. r  21 15 21 24

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

Solve and graph. –3y  15 –3 –7 –5 4 y  –5 7m < 21 7 3 –3 5 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

–2 –3 –3 Solve and graph. 1. h + 2 < 0 2 h  –2 2. c – 5 > –2 3 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

–2 Two-step Inequalities. Solve and graph. 1. -3 – 2n > 1 2 n  –2 2 n  –2 2. -4 – 2r < –6 r > 1 x 5 3. – 1 > 1 10 x  10 r 6 4. + 5 < 6 6 9 r < 6

-8 Two-step Inequalities. Solve and graph. 1. -1 + 2m > 15 8 n  8 8 n  8 2. -5k + 1 > 21 k < -4 -4 t 4 3. + 5 > 4 -4 t  -4 x 8 4. + 1 > 0 -8 x  -8