Lesson 5: Solving Inequalities in One Variable

Property

To solve an inequality:

1. We can add or subtract the same quantity on both sides.
2. We can multiply or divide both sides by the same positive number.
3. If we multiply or divide both sides by a negative number, we must reverse the direction of the inequality.

Examples

• To solve $2x - 5 < 9$, add 5 to both sides to get $2x < 14$. Then, divide by 2 to get $x < 7$. The inequality sign does not change.
• To solve $14 - 4x \geq 6$, subtract 14 to get $-4x \geq -8$. Then, divide by $-4$ and reverse the inequality sign to get $x \leq 2$.
• To solve $2 - \frac{x}{3} > 4$, subtract 2 to get $-\frac{x}{3} > 2$. Then, multiply by $-3$ and reverse the inequality sign to get $x < -6$.

Explanation

Solving an inequality is just like solving an equation, with one crucial exception. Remember this golden rule: if you multiply or divide both sides by a negative number, you MUST flip the direction of the inequality sign ($<$ becomes $>$, and vice versa).

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.