Solving One-Step and Two-Step Inequalities

Solving inequalities is a lot like solving regular equations. Your goal is still to isolate the variable on one side. However, there is one very important rule you must remember when working with inequalities.

The Golden Rule of Inequalities

When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

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

If you add, subtract, or multiply/divide by a positive number, the sign stays exactly the same.

Solving One-Step Inequalities

A one-step inequality requires only one operation to isolate the variable.

Example 1: Solve $x + 3 > 7$

Subtract $3$ from both sides: $x + 3 - 3 > 7 - 3$ $x > 4$

Example 2: Solve $-3x \geq 9$

Divide both sides by $-3$. Because we are dividing by a negative number, we must flip the $\geq$ sign to $\leq$: $\frac{-3x}{-3} \leq \frac{9}{-3}$ $x \leq -3$

Solving Two-Step Inequalities

Two-step inequalities require two operations. Always undo addition or subtraction first, then undo multiplication or division.

Example 3: Solve $2x - 1 < 5$

Step 1: Add $1$ to both sides. $2x - 1 + 1 < 5 + 1$ $2x < 6$

Step 2: Divide both sides by $2$. Since $2$ is positive, the sign stays the same. $\frac{2x}{2} < \frac{6}{2}$ $x < 3$

Graphing the Solutions

Once you have your solution, you can visualize it by graphing it on a number line:

• Open Circle ($\circ$): Use for $<$ (less than) or $>$ (greater than). This means the number itself is not included in the solution.
• Closed Circle ($\bullet$): Use for $\leq$ (less than or equal to) or $\geq$ (greater than or equal to). This means the number is included.
• Arrow Direction: If your variable is on the left side (like $x > 4$), the arrow points in the same direction as the inequality sign. For $>$, draw the arrow to the right. For $<$, draw the arrow to the left.