Lesson 6: Solving Inequalities Using Addition or Subtraction

Property

For any numbers $a$, $b$, and $c$, if $a < b$, then

For any numbers $a$, $b$, and $c$, if $a > b$, then

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.

Examples

• To solve $x + 7 < 15$, subtract 7 from both sides: $x + 7 - 7 < 15 - 7$, which simplifies to $x < 8$.
• To solve $y - 4 \geq -2$, add 4 to both sides: $y - 4 + 4 \geq -2 + 4$, which simplifies to $y \geq 2$.
• Given $12 > z + 5$, subtract 5 from both sides: $12 - 5 > z + 5 - 5$, so $7 > z$, which means $z < 7$.

Explanation

This property is just like it is for equations. You can add or subtract the same number on both sides of an inequality, and the relationship between the two sides stays the same. The inequality sign does not change.

Property

To solve an inequality using addition or subtraction:

1. We can add the same quantity to both sides of an inequality.
2. We can subtract the same quantity from both sides of an inequality.
3. The direction of the inequality sign remains unchanged when adding or subtracting.

Examples

Property

To solve linear inequality applications, translate the word problem into a mathematical inequality, then solve using algebraic properties.
The key steps are: identify the variable, write the inequality from the given constraints, solve algebraically, and interpret the solution in context.

Examples