Linear Inequality

Linear Inequality

Core Rule

When you multiply or divide both sides by a negative number, FLIP the inequality sign.

Solving Linear Inequalities

Treat like an equation, but watch for sign flips.

Example:

Compound Inequalities

AND (Intersection) — both must be true

Solution:

AND → the overlap of both solution sets

OR (Union) — at least one must be true

Solution:

OR → the combined solution sets

Absolute Value Inequalities

Less than (AND)

Example:

Greater than (OR)

Example:

Memory aid: Less thAND, GreatOR

Interval Notation

Inequality Interval Number Line
Open circle at , arrow right
Closed circle at , arrow right
Arrow left, open circle at
Closed at , open at

Inequalities in Two Variables

Symbol Line Shade
or Dashed (boundary not included) Above () or below ()
or Solid (boundary included) Above () or below ()

Test point method: Pick a point not on the line (usually ). Plug in:

  • If the inequality is true → shade the side containing the test point
  • If false → shade the other side

Word Problems with Inequalities

Look for keywords:
| Keyword | Symbol |
|---------|--------|
| at most, no more than, maximum | |
| at least, no less than, minimum | |
| fewer than, less than | |
| more than, greater than | |

Example:
"A student needs at least 80 points to pass. Each correct answer is worth 4 points. If the student already has 20 points, how many more correct answers are needed?"

SAT Tips

  1. The #1 mistake is forgetting to flip the sign — watch for multiplying/dividing by negatives
  2. Absolute value = distance from zero — means " is within 5 units of 3"
  3. When a problem says "which inequality represents...", test a value that should work and plug into each answer choice
  4. Systems of inequalities on the SAT usually ask for a point in the solution region — test each answer choice
  5. "No solution" for — absolute value is never negative, so has no solution
  6. "All real numbers" for — is always true
Select Banner