Compound Inequalities

What is a compound inequality A compound in equality is two inequalities joined together by either the word “and” or “or” X < 2 or x > 7 Read as: “x is less than 2 or x is greater than 7” X > 3 and x < 8 Read as: “x is greater than 3 and x is less than 8” However we usually see “and” statements like this 5 < x < 10 Read as: “x is greater than 5 and less than 10”

Solving an OR problem Just solve the left side and then the right side Graph both pieces on the number line 3x > 12 or -4x > 20 Divide by 3 divide by -4 x > 4 or x < -5 -5 4 The solutions are still where the arrows point.

Solving an AND problem 10 < 2x+4 < 28 You are trying to get x by itself -4 -4 -4 So subtract 4 from ALL sides 6 < 2x < 24 Now divide ALL sides by 2 2 2 3 < x < 12 The final answer is: x is greater than 3 and less than 12

Graphing AND Statements 3 < x < 12 3 12 To graph: put the two numbers on the number line. Draw open or closed circles. Connect the circles with a line segment (no arrows)

The hard part And statements always have to be written in this format: smaller # < x< bigger # The inequality arrows always face this direction in the end. The smaller number is always on the left The bigger number is always on the right

The hard part (continued) It becomes difficult when you divide by a negative. 8 < -2x ≤ 20 When you divide both sides by -2 -2 -2 -2 you have to flip the signs and get -4 > x ≥ -10 The problem is that this is no in the proper format so we have to switch everything back -10 ≤ x <-4 Note that the “same alligator” is eating the same number. I.e. Keep the equal sign with the same thing.

Practice Solve and Graph on a Number Line 2x + 5 > 13 or 3x + 5 < -13 14 ≤ 3x+5 ≤ 32

Answers to practice x > 4 or x < -6 -6 4 3 ≤ x ≤ 9 3 9