Solving Inequalities

Solving Inequalities

An inequality compares two expressions using <, >, \leq, or \geq instead of an equals sign. You solve inequalities almost exactly like equations — the same inverse operations apply — with one crucial exception.

The One Rule That's Different

When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is because multiplying by a negative number reverses the relative order of numbers on the number line — for example, 2 < 5, but multiplying both sides by -1 gives -2 > -5, not -2 < -5.

Adding or subtracting any number (positive or negative) never flips the sign — only multiplying or dividing by a negative does.

Example

Solve -3x + 4 \leq 19.

1. Subtract 4 from both sides (no sign flip, since this is subtraction): -3x \leq 15
2. Divide both sides by -3. Since we're dividing by a negative number, flip the inequality sign: x \geq -5

So the solution is all x greater than or equal to -5.

Graphing on a Number Line

To graph x \geq -5: draw a number line, place a closed (filled-in) circle at -5 because -5 itself is included (the inequality is \geq, not strict), and shade the ray extending to the right, toward larger numbers. For a strict inequality like x < 3, you would instead use an open circle at 3 (since 3 is not included) and shade to the left, toward smaller numbers.

• Closed/filled circle: used for \leq or \geq (the boundary point is included).
• Open circle: used for < or > (the boundary point is excluded).
• Shade in the direction of the values that make the inequality true.

Multi-Step Inequalities

Just like equations, some inequalities require distributing or combining like terms before you can isolate the variable. For example, to solve 2(x - 3) > x + 5, first distribute the left side to get 2x - 6 > x + 5, then subtract x from both sides to get x - 6 > 5, and finally add 6 to both sides to get x > 11. None of these steps involved multiplying or dividing by a negative number, so the inequality symbol never had to flip.

Key Takeaways

• Solve inequalities the same way as equations, using inverse operations.
• Flip the inequality sign only when multiplying or dividing both sides by a negative number.
• Use a closed circle for \leq / \geq and an open circle for < / > when graphing.
• Shade the number line in the direction of the solutions.