Lesson 4: Solving Multi-Step Inequalities

Property

When clearing fractions from an inequality, multiply every term on both sides by the least common denominator (LCD). If the LCD is negative, reverse the inequality symbol.

Examples

Property

To solve a multi-step linear inequality, follow a systematic flow:

1. Simplify each side completely (distribute and combine like terms).
2. Use the Addition or Subtraction Properties of Inequality to collect all variable terms on one side and all constant terms on the other side.
3. Use the Multiplication or Division Properties of Inequality to isolate the variable. (Remember to reverse the inequality sign if you multiply or divide by a negative number!)

Examples

• Example 1: Solve $3x + 5 > 20$.

Subtract 5 from both sides to get $3x > 15$.
Divide by 3 to get $x > 5$.

• Example 2 (Variables on both sides): Solve $7p - 2 \leq 3p + 10$.

Subtract $3p$ from both sides to gather variables on the left: $4p - 2 \leq 10$.
Add 2 to both sides to gather constants on the right: $4p \leq 12$.
Divide by 4 to get $p \leq 3$.

• Example 3 (Negative division): Solve $5(k - 2) > -20$.

Distribute to get $5k - 10 > -20$.
Add 10 to both sides: $5k > -10$.
Divide by 5 to get $k > -2$. (The sign stays the same because we divided by a positive 5).

Explanation

Solving a multi-step inequality uses the exact same strategy as solving a multi-step equation: clean up both sides, move the letters to one team and the numbers to the other, and then isolate the variable. The only difference is the golden rule of inequalities—you must stay highly alert during the very last step. If you divide or multiply by a negative number to get the variable by itself, you must flip the inequality symbol.

Property

To solve inequalities with variables on both sides:
(1) Add or subtract variable terms to collect all variables on one side,
(2) Add or subtract constants to collect all constants on the other side,
(3) Use multiplication or division to isolate the variable, remembering to reverse the inequality symbol when multiplying or dividing by a negative number.

Examples