Compound Inequalities

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.

Definition

And is the Intersection

Or is the Union

3 x + 2 < 14 and 2 x – 5 > –11

2 x + 7 < –11 or –3 x – 2 < 13

x < -3 or x > 2

Disjunction

1. Graph each inequality .

_l_____l______l_____l_____l______l______l_____l_____-4 -3 -2 -1 0 1 2 3 4

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.

x > 1 and x < 3 ConjunctionGraph Each Inequality

l l l l l l l l l l -1 0 1 2 3 4 5 6

And means intersection, the solution includes what the inequalitieshave in common (the overlap)

l l l l l l l l l l -1 0 1 2 3 4 5 6{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.

Solving A Compound InequalityExample 1: Solve 3x + 1< 4 or 2x – 5 > 7

1. Solve each inequality:3x + 1 < 4 2x – 5 > 7

x < 1 x > 6

Graph the solution {x: x <1 or x > 6}

l l l l l l l l l l l l l -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Solving A Compound Inequality

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 2 3 4 5 6 7 8

Your TurnSolve the compound inequality.-2 < 2x + 6 < 12

Solve the inequalitiesGraph the solutionsDetermine the final solutionExpress your solution in set notation

SOLUTION-2 < 2x + 6 < 12

-2 < 2x + 6-6 - 6-8 < 2x 2 2-4 < x x > -4

2X + 6 < 12 - 6 - 62X < 6 2 2 x < 3

l l l l l l l l l -4 -3 -2 -1 0 1 2 3 4

Your Turn2x + 3 < 1 or 3x – 5 > 4

2x + 3 < 1 or 3x – 5 > 4

Solution

2x + 3 < 1 - 3 -32x < -2 2 2 x < -1

3x – 5 > 4 + 5 +53x > 93 3x > 3

l l l l l l l l l -4 -3 -2 -1 0 1 2 3 4

You Did It!!!!!