Solving Linear Inequalities and Compound Inequalities.

Solving Linear Inequalitiesand

Compound Inequalities

Inequality SymbolsInequality Symbols

is less

than

is greater than

is less than or equal to

is greater than or equal to

is not equal to

Linear Inequality

• Can be written in the form ax+b<0, ax+b>0, ax+b≤0, or ax+b≥0 where a and b are real numbers and a≠0

• Has one variable to the first power.

for example: 2x-3<8

• A solution is a value of the variable that makes the inequality true.

x could equal -3, 0, 1, etc.

Transformations for InequalitiesTransformations for Inequalities

• Add/subtract the same number on each side of an inequality

• Multiply/divide by the same positive number on each side of an inequality

• If you multiply or divide by a negative number, you MUST flip the inequality sign!

Ex: Solve the inequality.

2x-3<8

+3 +3

2x<11

2 2

x<2

11

1373 x

63 x

2x

Flip the sign after dividing by the -3!

Graphing Linear Inequalities

• Remember:

< and > signs will have an open circle

and signs will have a closed circle

graph of graph of

2

11x 2x

4 5 6 7 -3 -2 -1 0

Example: Solve and graph the solution.

121097 xx

1239 xx321x7

6 7 8 9

Example: Solve and graph the solution.

6( 5 3 ) 3(6 10)p p 30 18 18 30p p

18p

30 30

18p

This is a true statement, therefore the solution is ALL REAL NUMBERS.

Compound Inequality

• An inequality joined by “and” or “or”.

Examples

“and”/intersection “or”/union

think between think oars on a boat

13 x

-4 -3 -2 -1 0 1 2

4or 2 xx

-3 -2 -1 0 1 2 3 4 5

Example: Solve & graph.

-9 < t+4 < 10

-4 -4 -4

-13 < t < 6

Think between!

-13 6

Solve & graph. -6x+9 < 3 or -3x-8 > 13

-6x < -6 -3x > 21

x > 1 or x < -7

Flip signs

Think oars

-7 1