Multi-Step and Compound Inequalities

Solving multi-step inequalities is very similar to solving multi-step equations. You will use the exact same algebraic techniques: distributing, combining like terms, and moving variables to isolate your unknown. However, there is one critical rule you must always remember when dealing with inequalities.

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.

• $<$ becomes $>$
• $\leq$ becomes $\geq$

If you are just adding or subtracting, or if you multiply/divide by a positive number, the sign stays exactly the same.

Steps to Solve Multi-Step Inequalities

1. Distribute to remove any parentheses.
2. Combine like terms on each side of the inequality.
3. Move all variables to one side (usually the left) by adding or subtracting.
4. Isolate the variable using addition, subtraction, multiplication, or division. (Remember the Golden Rule!)

Example Problems

Example 1: Variables on Both Sides with Distribution

Solve: $3(x - 2) > x + 4$

Step 1: Distribute the $3$. $3x - 6 > x + 4$

Step 2: Subtract $x$ from both sides to group the variables. $2x - 6 > 4$

Step 3: Add $6$ to both sides. $2x > 10$

Step 4: Divide by $2$. Since $2$ is positive, keep the sign the same. $x > 5$

Example 2: Flipping the Inequality Sign

Solve: $-2x + 5 \leq 3x - 10$

Step 1: Subtract $3x$ from both sides. $-5x + 5 \leq -10$

Step 2: Subtract $5$ from both sides. $-5x \leq -15$

Step 3: Divide by $-5$. Since we are dividing by a negative number, flip the sign! $x \geq 3$

Example 3: Combining Like Terms

Solve: $2(x + 3) - x \geq 7$

Step 1: Distribute the $2$. $2x + 6 - x \geq 7$

Step 2: Combine like terms ($2x - x$). $x + 6 \geq 7$

Step 3: Subtract $6$ from both sides. $x \geq 1$

A Note on Compound Inequalities

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

• AND inequalities (e.g., $-2 < x \leq 5$) mean the solution must satisfy both conditions. The solution is the overlapping region between the two.
• OR inequalities (e.g., $x < -1$ or $x > 3$) mean the solution satisfies at least one condition. The graph typically points outward in opposite directions.

You solve compound inequalities using the exact same multi-step rules, just applied to multiple parts simultaneously!