Introduction

Multi-step inequalities extend the ideas from one-step inequalities by adding layers: combining like terms, distributing, and sometimes dealing with negative coefficients.
Because inequalities describe ranges of solutions rather than single values, each step must preserve the direction of the inequality — except when multiplying or dividing by a negative number, which flips the inequality sign.

This article assumes you already understand:

• How to isolate a variable in a one-step inequality
• How to add, subtract, multiply, or divide both sides of an inequality
• The meaning of inequality symbols ($<, >, \le, \ge$)

What Are Multi-Step Inequalities?

A multi-step inequality is any inequality that requires more than one algebraic operation to isolate the variable.

Examples:

• $3x - 7 \le 11$
• $4 - 2(x + 3) > 10$
• $5x + 8 \ge 3x - 4$

Typical steps include:

• Combining like terms
• Using the distributive property
• Moving variable terms to one side
• Handling negative coefficients carefully

Key Idea: Flipping the Inequality

When solving inequalities, the only time the inequality sign flips is when you:

• Multiply both sides by a negative number
• Divide both sides by a negative number

Examples:

• If $-2x > 6$, dividing by $-2$ gives $x < -3$
• If $-5y \le 20$, dividing by $-5$ gives $y \ge -4$

This rule is essential for solving inequalities with negative coefficients.

General Strategy for Solving Multi-Step Inequalities

• 1. Distribute
  Apply the distributive property if parentheses are present.
• 2. Combine like terms
  Simplify each side of the inequality.
• 3. Move variable terms to one side
  Use addition or subtraction to gather all $x$-terms on one side.
• 4. Isolate the variable
  Multiply or divide to solve for the variable.
  Flip the sign if dividing/multiplying by a negative.
• 5. Represent the solution
  • As an inequality
  • On a number line
  • In interval notation (optional)

Worked Examples

Example 1

Solve: $$3x - 5 \ge 7$$ Steps:

• Add $5$ to both sides: $3x \ge 12$
• Divide by $3$: $x \ge 4$

Example 2

Solve: $$4 - 2(x + 1) < 6$$ Steps:

• Distribute: $4 - 2x - 2 < 6$
• Combine like terms: $2 - 2x < 6$
• Subtract $2$: $-2x < 4$
• Divide by $-2$ and flip the sign: $x > -2$

Example 3

Solve: $$5x + 8 \le 3x - 4$$ Steps:

• Subtract $3x$: $2x + 8 \le -4$
• Subtract $8$: $2x \le -12$
• Divide by $2$: $x \le -6$

Calculator

Exercises

1. Solve the inequality: $4x - 7 > 9$

   Solution

   $4x - 7 > 9$
   Add $7$: $4x > 16$
   Divide by $4$:
   $x > 4$
2. Solve: $3 - 2(x - 4) \le 11$

   Solution

   $3 - 2(x - 4) \le 11$
   Distribute: $3 - 2x + 8 \le 11$
   Combine: $11 - 2x \le 11$
   Subtract $11$: $-2x \le 0$
   Divide by $-2$ (flip sign):
   $x \ge 0$
3. Solve: $5x + 12 \ge 2x - 6$

   Solution

   $5x + 12 \ge 2x - 6$
   Subtract $2x$: $3x + 12 \ge -6$
   Subtract $12$: $3x \ge -18$
   Divide by $3$:
   $x \ge -6$
4. Solve the inequality with a negative coefficient: $-3x + 5 < 2$

   Solution

   $-3x + 5 < 2$
   Subtract $5$: $-3x < -3$
   Divide by $-3$ (flip sign):
   $x > 1$
5. Solve: $7 - (x + 2) > 3x$

   Solution

   $7 - (x + 2) > 3x$
   Distribute: $7 - x - 2 > 3x$
   Combine: $5 - x > 3x$
   Add $x$: $5 > 4x$
   Divide by $4$:
   $x < \frac{5}{4}$
6. Solve: $-4(2x - 1) \ge 12$

   Solution

   $-4(2x - 1) \ge 12$
   Distribute: $-8x + 4 \ge 12$
   Subtract $4$: $-8x \ge 8$
   Divide by $-8$ (flip sign):
   $x \le -1$
7. Solve: $2x + 3 < 5x - 9$

   Solution

   $2x + 3 < 5x - 9$
   Subtract $2x$: $3 < 3x - 9$
   Add $9$: $12 < 3x$
   Divide by $3$:
   $4 < x$ or $x > 4$
8. Solve the inequality and remember to flip the sign: $-5(x - 2) \le 15$

   Solution

   $-5(x - 2) \le 15$
   Distribute: $-5x + 10 \le 15$
   Subtract $10$: $-5x \le 5$
   Divide by $-5$ (flip sign):
   $x \ge -1$