Compound Inequalities A compound inequality is a sentence with two inequality statements joined either by the word “or” or by the word “and.” “And” indicates that both statements of the compound sentence are true at the same time. It is the overlap or intersection of the solution sets for the individual statements. “Or” indicates that, as long as either statement is true, the entire compound sentence is true. It is the combination or union of the solution sets for the individual statements.

3 x + 2 –11 2 x + 7 < –11 or –3 x – 2 < 13

1. Graph each inequality. _l_____l______l_____l_____l______l______l_____l_____ Everything that is mentioned in the two inequalities is a solution. The set of all “x”s such that x is less than -3 or x is greater than 2.

Graph Each Inequality l l l l l And means intersection, the solution includes what the inequalities have in common (the overlap) l l l l l {a: 1 < a < 3 } The set of all “a”s such that 1 is less than or equal to a which is less than or equal to 3.

Example 1: Solve 3x Solve each inequality: 3x x 6 Graph the solution {x: x 6} l l l l l l l l l l l l l

This compound inequality can be interpreted as 1 < 2c – 7 < 7 1 < 2c – 7 and 2c – 7 < 7 c > 4 c < 7 l l l l l l l

Solve the compound inequality. -2 < 2x + 6 < 12 Solve the inequalities Graph the solutions Determine the final solution Express your solution in set notation

-2 < 2x + 6 < < 2x < 2x < x x > -4 2X + 6 < X < x < 3 l l l l l l l l l

2x + 3 4

2x + 3 < x < x < -1 3x – 5 > x > 9 3 x > 3 l l l l l l l l l