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Linear Inequalities: Solving, Graphing, and the Flip-the-Sign Rule

Master linear inequalities: solving them just like equations, learning the flip-the-sign rule, and graphing them on a number line or coordinate plane.

Practice Linear Inequalities

What Are Linear Inequalities?

A linear inequality works exactly like a linear equation, except instead of an equals sign, you have a <, >, ≤, or ≥. Instead of a single answer like x = 5, the solution is a whole range of numbers, like x > 5.

Solving them: 99% the same as equations

To solve an inequality, do the exact same steps you'd do to isolate x in an equation: add, subtract, multiply, and divide on both sides.

The ONLY difference (The Golden Rule of Inequalities):

If you multiply or divide both sides by a negative number, you MUST flip the inequality sign.

Worked Examples

Example 1: Solving without a flip

Easy

Solve and graph: 2x + 3 > 8

Step 1: Subtract 3 from both sides
2x > 5
Isolate the variable term.
Step 2: Divide by 2
x > 2.5
Since we divided by a positive number, the inequality sign stays exactly the same.
Step 3: Graph on a number line
Open circle at 2.5, arrow to the right
Use an open circle because it's "greater than" (not equal to), and shade to the right.
Final Answer:
x > 2.5

Example 2: The Flip Rule

Medium

Solve and graph: -3x + 4 ≤ 16

Step 1: Subtract 4 from both sides
-3x ≤ 12
Isolate the -3x.
Step 2: Divide by -3
x ≥ -4
CRITICAL STEP: We divided by a negative number, so we MUST flip the ≤ to ≥.
Step 3: Graph on a number line
Closed circle at -4, arrow to the right
Use a closed circle because of the "or equal to", and shade the values greater than -4.
Final Answer:
x ≥ -4

Graphing on a Number Line

  • Open circle: for < or > (the number isn't included).
  • Closed circle: for ≤ or ≥ (the number is included).
  • Shading: shade in the direction the inequality points (if x is on the left!). E.g., for x > 5, shade to the right.

Graphing on a Coordinate Plane

If the inequality has both x and y (like y < 2x + 1), you graph the boundary line just like a regular equation.

  • Use a dashed line for < or >.
  • Use a solid line for ≤ or ≥.
  • Then, test a point (like (0,0)) to see which side of the line makes the inequality true, and shade that entire half of the graph.

Common mistakes to avoid

  • Flipping when subtracting. You only flip the sign when you multiply or divide by a negative, not when you subtract a negative.
  • Flipping when dividing a negative by a positive. The number you are dividing by dictates the flip, not the number being divided.
  • Shading the wrong direction. It's safest to read from the variable. 4 < x means the same as x > 4 — it helps to rewrite it with x on the left before graphing.

Where this shows up next

Inequalities naturally lead to systems of inequalities, absolute value inequalities, and eventually determining the domain of more complex functions (where square roots or denominators restrict possible values).

Continue Your Algebra Journey

Mastered linear inequalities? Move on to quadratics.

Solving Inequalities Worksheet

Free printable practice: 25 problems with number-line answers.

Quadratic Equations

Factoring, the quadratic formula, and parabolas.

Master the Flip-the-Sign Rule

Practice solving and graphing inequalities with guided feedback.

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