What is the difference between and and or? AND means intersection -what do the two items have in common? OR means union -if it is in one item, it is in the solution A B

● ● ● 1) Graph x < 4 and x ≥ 2 a) Graph x < 4 o o b) Graph x ≥ 2 3 4 2 o 3 4 2 o b) Graph x ≥ 2 3 4 2 ● ● c) Combine the graphs d) Where do they intersect? ● 3 4 2 o

● ● 2) Graph x < 2 or x ≥ 4 a) Graph x < 2 o o b) Graph x ≥ 4 3 4 2 o 3 4 2 o b) Graph x ≥ 4 3 4 2 ● 3 4 2 ● c) Combine the graphs

3) Which inequalities describe the following graph? -2 -1 -3 o y > -3 or y < -1 y > -3 and y < -1 y ≤ -3 or y ≥ -1 y ≥ -3 and y ≤ -1 Answer Now

4) Graph the compound inequality 6 < m < 8 When written this way, it is the same thing as 6 < m AND m < 8 It can be rewritten as m > 6 and m < 8 and graphed as previously shown, however, it is easier to graph everything between 6 and 8! 7 8 6 o

5) Which is equivalent to -3 < y < 5? y > -3 or y < 5 y > -3 and y < 5 y < -3 or y > 5 y < -3 and y > 5 Answer Now

6) Which is equivalent to x > -5 and x ≤ 1? Answer Now

● ● 7) 2x < -6 and 3x ≥ 12 o o o o Solve each inequality for x Graph each inequality Combine the graphs Where do they intersect? They do not! x cannot be greater than or equal to 4 and less than -3 No Solution!! -3 -6 o -3 -6 o 4 7 1 o ● 4 7 1 o ●

8) Graph 3 < 2m – 1 < 9 Remember, when written like this, it is an AND problem! 3 < 2m – 1 AND 2m – 1 < 9 Solve each inequality. Graph the intersection of 2 < m and m < 5. 5 -

9) Graph x < 2 or x ≥ 4 5 -

The whole line is shaded!! 10) Graph x ≥ -1 or x ≤ 3 The whole line is shaded!!

Solving a Compound Inequality with And Example 3 Solving a Compound Inequality with And Solve -5 ≤ 2x + 3 < 7. Then graph the solution. Isolate the variable between the inequality symbols. –5 ≤ 2x + 3 < 7 -3 -3 -3 Subtract 3 from all three sides –8 ≤ 2x < 4 2 2 2 Divide each side by 2. –4 ≤ x < 2 The solution is all real numbers that are greater than or equal to -4 and less than 2. -5 -4 -3 -2 -1 0 1 2 3

Solving a Compound Inequality with Or Solve x + 5 < -6 or 3x > 12 Solve each part separately. or 3x >12 x + 5 ≤ –6 -5 -5 3 3 x ≤ –11 or x > 4 -12 -10 -8 -6 -4 -2 0 2 4 6

Solve –3 < –1 – 2x ≤ 5. Then graph the solution. Example 5 Reversing the sign Solve –3 < –1 – 2x ≤ 5. Then graph the solution. +1 +1 +1 –2 < – 2x ≤ 6 Reverse the inequalities when you divide by a negative –2 –2 –2 1 > x ≥ –3 -4 -3 -2 -1 0 1 2

Try these Solve and graph the inequality. 1. 2. 3. 4. -2 -1 0 1 2 3 4 5 6 2. -2 -1 0 1 2 3 4 5 6 3. -4 -3 -2 -1 0 1 2 3 4 4. -10 -9 -8 -7 -6 -5 -4 -3 -2