This document discusses solving linear inequalities. It defines inequality symbols and notes that a linear inequality involves one variable to the first power. It explains that a solution makes the inequality true. Transformations can be made to inequalities, such as adding/subtracting the same number or multiplying/dividing by a positive number, but multiplying/dividing by a negative number requires flipping the inequality sign. Graphing linear inequalities involves open or closed dots depending on the inequality sign. Compound inequalities joined by "and" or "or" can be solved by thinking of the solutions between or as oars on a boat.

1. Solving Linear Inequalities

2. Inequality SymbolsInequality Symbols
≠≥≤><
Less
than
Greater
than
Less
than or
equal to
Greater
than or
equal to
Not
equal to

3. Linear Inequality
• Inequality with 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.

4. 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!

5. Ex: Solve the inequality.
2x-3<8
+3 +3
2x<11
2 2
x<
2
11
1373 ≤+− x
63 ≤− x
2−≥x
Flip the sign after
dividing by the -3!

6. Graphing Linear Inequalities
• Remember:
< and > signs will have an open dot o
and signs will have a closed dot
graph of graph of
≤ ≥
2
11
<x 2−≥x
•
4 5 6 7 -3 -2 -1 0

7. Example: Solve and graph the solution.
121097 −≥+ xx
1239 −≥ x
x321≥
x≥7
6 7 8 9

8. Compound Inequality
• An inequality joined by “and” or “or”.
Examples
“and” “or”
think between
think oars on a boat
13 ≤<− x
-4 -3 -2 -1 0 1 2
4or2 >−≤ xx
-3 -2 -1 0 1 2 3 4 5

9. Example: Solve & graph.
-9 < t+4 < 10
-13 < t < 6
Think between!
-13 6

10. Last example! Solve & graph.
-6x+9 < 3 or -3x-8 > 13
-6x < -6 -3x > 21
x > 1 or x < -7
Flip signs
Think oars
-7 1