Introduction

• This article builds on your understanding of inequality symbols ($<, >, \le, \ge$).
• Our goal is to solve one-step inequalities using inverse operations, just like with one-step equations.
• The key difference: solutions to inequalities are sets of numbers, not just one number.

What Is a One-Step Inequality?

• A one-step inequality is an inequality where the variable can be isolated using one inverse operation.
• Examples:
  • $x + 5 > 12$
  • $x - 3 \le 10$
  • $4x > 20$
  • $\frac{x}{3} < 7$

Inverse Operations Refresher

To isolate the variable, we undo the operation being applied to it.

• Addition ↔ Subtraction
• Multiplication ↔ Division

Examples:

• To undo $+5$, subtract $5$.
• To undo $\times 4$, divide by $4$.

Solving Addition and Subtraction Inequalities

Example 1: $x + 7 > 10$

• Subtract $7$ from both sides.
• $x > 3$

Example 2: $x - 4 \le 9$

• Add $4$ to both sides.
• $x \le 13$

Important: Adding or subtracting the same number on both sides does not change the direction of the inequality.

Solving Multiplication and Division Inequalities

Example 1: $4x < 20$

• Divide both sides by $4$.
• $x < 5$

Example 2: $\frac{x}{3} \ge 2$

• Multiply both sides by $3$.
• $x \ge 6$

Important rule:

• When you multiply or divide by a negative number, the inequality reverses direction.

Example 3: $-2x > 8$

• Divide both sides by $-2$.
• Flip the inequality: $x < -4$

Common Mistakes to Avoid

• Forgetting to flip the inequality when multiplying or dividing by a negative.
• Treating inequalities like equations and giving only one number.
• Forgetting that solutions represent all numbers that make the inequality true.

Calculator

Exercises

1. Solve the inequality: $x + 9 < 20$

   Solution

   $x < 11$
   Subtract $9$ from both sides.
2. Solve: $x - 6 \ge 4$

   Solution

   $x \ge 10$
   Add $6$ to both sides.
3. Solve: $5x > 30$

   Solution

   $x > 6$
   Divide both sides by $5$.
4. Solve: $\frac{x}{4} \le 3$

   Solution

   $x \le 12$
   Multiply both sides by $4$.
5. Solve the inequality and state whether the inequality flips: $-3x < 12$

   Solution

   $x > -4$
   Divide by $-3$ and flip the inequality.
6. Graph the solution to $x > -2$ on a number line.

   Solution

   Open circle at $-2$, shade to the right.
7. True or false: Solving $-x > 5$ requires flipping the inequality.

   Solution

   True — dividing by $-1$ flips the inequality.
8. Solve: $2 + x \le 10$

   Solution

   $x \le 8$
   Subtract $2$ from both sides.