Linear and Compound Inequalities

An inequality compares two values, showing if one is less than, greater than, or simply not equal to another. Solving inequalities is very similar to solving regular equations, but with a few special rules.

Solving Linear Inequalities

To solve a linear inequality, your goal is to isolate the variable on one side. You can add or subtract numbers from both sides just like you do with equations.

The Golden Rule of Inequalities: Whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign (e.g., change $>$ to $<$).

Example: Solve $-3x + 7 > 1$

1. Subtract $7$ from both sides: $-3x > 1 - 7$ $-3x > -6$
2. Divide both sides by $-3$. Because you are dividing by a negative number, remember to flip the $>$ sign to $<$: $x < \frac{-6}{-3}$ $x < 2$

Compound Inequalities

A compound inequality consists of two separate inequalities joined by either "and" or "or".

• "And" (Intersection): The solution must make both inequalities true at the same time. These are often written in a condensed form, like $a < x < b$.
• "Or" (Union): The solution must make at least one of the inequalities true.

Example: Solve the "and" compound inequality $2 < 3x - 1 \leq 8$

To solve a condensed "and" inequality, whatever operation you do to the middle, you must do to all three parts (left, middle, and right) simultaneously.

1. Add $1$ to all three parts to isolate the $x$ term in the middle: $2 + 1 < 3x - 1 + 1 \leq 8 + 1$ $3 < 3x \leq 9$
2. Divide all three parts by $3$: $\frac{3}{3} < \frac{3x}{3} \leq \frac{9}{3}$ $1 < x \leq 3$

The solution is any number strictly greater than $1$ and less than or equal to $3$.