Solving One- & Two-Step Inequalities

Q: How do I graph the answer?

On a number line: open circle for $ $, closed circle for $\leq$ or $\geq$, then shade toward the solutions. $x > -3$ is an open circle at $-3$ with a ray pointing right.

Solving an inequality like $x + 5 < 9$ or $-3x \leq 12$ works almost exactly like solving an equation: use inverse operations on both sides until the variable stands alone.

There is exactly one new rule, and it's the one this whole topic hangs on: when you multiply or divide both sides by a negative number, flip the inequality sign.

Solve it like an equation

Adding or subtracting works with no surprises. To solve $x + 5 < 9$ , subtract $5$ from both sides: $x < 4$ . Every number less than $4$ is a solution.

Multiplying or dividing by a positive number is also safe. Dividing $4x \leq -24$ by $4$ gives $x \leq -6$ , and the sign keeps pointing the same way.

The flip rule

When both sides get multiplied or divided by a negative number, the inequality sign reverses direction. Dividing $-3x \leq 12$ by $-3$ gives $x \geq -4$ — the $\leq$ flipped to $\geq$ .

Why? Multiplying by a negative reflects every number across zero, which reverses their order. $2 < 5$ is true, but multiply both sides by $-1$ and you have $-2$ and $-5$ — and $-2 > -5$ . Flipping the sign keeps the statement true.

The rule is only about multiplying and dividing. Adding or subtracting a negative number never flips the sign.

Two-step inequalities

Use the same order as two-step equations: move the constant first, then deal with the coefficient — flipping only if that last step multiplies or divides by a negative number.

For $2x + 7 > 1$ : subtract $7$ to get $2x > -6$ , then divide by the positive $2$ to get $x > -3$ . No flip, because the divisor was positive — the negative on the other side doesn't matter.

Worked examples

Example 1: one step, no flip

Solve $x + 5 < 9$ .

Start with the inequality

x + 5 < 9

Subtract

5

from both sides

x < 4

Check with a test number:

3 + 5 = 8 < 9

✓

Answer: $x < 4$

Example 2: divide by a negative

Solve $-3x \leq 12$ .

Start with the inequality

-3x \leq 12

Divide both sides by

-3

and flip the sign

x \geq -4

Check with a test number:

-3(0) = 0 \leq 12

✓

Answer: $x \geq -4$

Example 3: two steps

Solve $2x + 7 > 1$ .

Start with the inequality

2x + 7 > 1

Subtract

7

from both sides

2x > -6

Divide both sides by

2

— positive, so no flip

x > -3

Answer: $x > -3$

Try one yourself

Common questions

Why does the sign flip when I multiply or divide by a negative?

Multiplying by a negative reflects numbers across zero, and that reverses their order — the bigger number becomes the smaller one. $2 < 5$ turns into $-2 > -5$ . Flipping the sign is what keeps the statement true.

Does adding or subtracting a negative number flip the sign?

No. Only multiplying or dividing both sides by a negative number flips it. Solving $x - 3 > 2$ by adding $3$ keeps the sign: $x > 5$ .

How do I check my answer?

Pick an easy test number from your answer and plug it into the original inequality. For $x \geq -4$ , try $0$ : $-3(0) = 0 \leq 12$ is true, so the answer checks. Also test a number outside your answer — it should fail.

How do I graph the answer?

On a number line: open circle for $<$ or $>$ , closed circle for $\leq$ or $\geq$ , then shade toward the solutions. $x > -3$ is an open circle at $-3$ with a ray pointing right.

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Solving One- & Two-Step Inequalities

Solve it like an equation

The flip rule

Two-step inequalities

Worked examples

Example 1: one step, no flip

Example 2: divide by a negative

Example 3: two steps

Try one yourself

Common questions

Keep learning

Graphing Inequalities on a Number Line

Two-Step Inequalities

Two-Step Equations

Variables on Both Sides