Linear Inequalities

General Tips

All the same math rules apply—with one big caveat: Solve inequalities just like equations (distribute, combine terms, isolate the variable). But if you multiply or divide both sides by a negative, flip the inequality sign.
Why do you flip the sign after multiplying/dividing by a negative?

Start with a true inequality chain:

$$1 < 2 < 3$$

Now multiply all parts by $-1$:

$$-1 < -2 < -3$$

This is not true on the number line.

To fix it, flip the inequality signs:

$$-1 > -2 > -3$$

Translate phrases into inequality symbols:

> (Greater Than)	< (Less Than)	≥ (Greater Than or Equal To)	≤ (Less Than or Equal To)
greater than more than exceeds above higher than larger than	less than fewer than below under is smaller than	at least no less than not less than minimum of minimum reuired must be at least as few as at or above	at most no more than not more than maximum of must be at most as many as a maximum of a ceiling of at or below up to and including

How to graph linear inequalities:
- Inclusive inequalities (≤ or ≥): use a solid line; points on the line are solutions.
- Strict inequalities (< or >): use a dashed line; points on the line are not solutions.
- Shade above the line for “greater than” and below for “less than.”
- Pick 1 point on the line: move up for greater than, down for less than. That's your shaded area.
Compound inequalities:
- AND inequalities: bounded on both sides (e.g., 3 < x ≤ 7).
- OR inequalities: disjoint ranges, often from absolute value expressions.

Systems of inequalities use more than one constraint. To solve, substitute one constraint into the other.
Graph inequalities correctly: solid vs. dashed line, and shading direction.
Rewrite equations to slope-intercept form for easy comparison.
Plug in coordinate choices to verify solutions, especially when given tables.