One-Step Inequalities

Q: Why does the sign flip when I divide by a negative?

Because negatives reverse order. $2 -6$. Every multiply or divide by a negative swaps which side is bigger, so the symbol must swap with it.

Q: What about $-x > 3$, where the coefficient is just a negative sign?

That's really $-1 \cdot x > 3$. Divide both sides by $-1$ and flip: $x < -3$. Any time the variable ends up with a negative in front, one more divide-and-flip finishes the problem.

Q: How do I check my answer?

Pick a simple number that your answer says should work — $0$ is great when it's in the region — and substitute it into the original inequality. True statement, correct answer. If it comes out false, you almost certainly missed a flip.

A one-step inequality is solved exactly like a one-step equation: do the opposite operation to both sides to get the variable alone. If the inequality adds $5$ , subtract $5$ from both sides. If it multiplies by $3$ , divide both sides by $3$ .

There is exactly one new rule, and it's the rule this whole unit turns on: when you multiply or divide both sides by a negative number, you must flip the inequality sign. Miss that flip and every negative-coefficient problem comes out backwards.

Solve it like an equation

Adding or subtracting the same number on both sides never changes the direction of an inequality. If $x + 5 > 12$ , subtracting $5$ from both sides gives $x > 7$ — same symbol, done in one move.

Multiplying or dividing both sides by a positive number is also safe. $3x \leq 15$ becomes $x \leq 5$ after dividing by $3$ , and the symbol stays put.

The sign-flip rule

When you multiply or divide both sides by a negative number, flip the inequality sign: $<$ becomes $>$ , and $\geq$ becomes $\leq$ .

Here's why. Start with a true statement: $2 < 6$ . Multiply both sides by $-1$ and you get $-2$ and $-6$ — but $-2$ is actually greater than $-6$ . Multiplying by a negative reverses the order of every pair of numbers, so the symbol has to reverse too, or the statement turns false.

The flip happens only for multiplying and dividing by a negative. Adding or subtracting a negative number never flips anything, and neither does dividing by a positive.

Graph the solution

A solved inequality like $x > 7$ graphs on a number line the usual way: an open circle at $7$ for a strict symbol ( $>$ or $<$ ), a closed circle for $\geq$ or $\leq$ , and shading toward the solutions — right for greater than, left for less than.

A quick check catches flip mistakes: pick an easy number from your shaded region and substitute it into the original inequality. If it makes a true statement, your answer is right.

For example, $x \geq -1$ graphs as a closed circle at $-1$ (the boundary is included) with the ray shading right toward the larger numbers.

Worked examples

Example 1: subtraction, no flip

Solve $x + 9 > 14$ .

Start with the inequality

x + 9 > 14

Subtract

9

from both sides

x > 5

Check with

x = 6

6 + 9 = 15 > 14

✓

Answer: $x > 5$

Example 2: division by a positive, no flip

Solve $4m \leq 28$ .

Start with the inequality

4m \leq 28

Divide both sides by

4

— positive, so the sign stays

m \leq 7

Answer: $m \leq 7$

Example 3: division by a negative — flip

Solve $-3y < 12$ .

Start with the inequality

-3y < 12

Divide both sides by

-3

and flip the sign

y > -4

Check with

y = 0

-3(0) = 0 < 12

✓

Answer: $y > -4$

Try one yourself

Common questions

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

Because negatives reverse order. $2 < 6$ is true, but after multiplying both sides by $-1$ you have $-2$ and $-6$ , and $-2 > -6$ . Every multiply or divide by a negative swaps which side is bigger, so the symbol must swap with it.

Does subtracting a number ever flip the sign?

No. Adding or subtracting anything — positive or negative — slides both sides the same distance and keeps their order. Only multiplying or dividing by a negative flips the sign.

What about

-x > 3

, where the coefficient is just a negative sign?

That's really $-1 \cdot x > 3$ . Divide both sides by $-1$ and flip: $x < -3$ . Any time the variable ends up with a negative in front, one more divide-and-flip finishes the problem.

How do I check my answer?

Pick a simple number that your answer says should work — $0$ is great when it's in the region — and substitute it into the original inequality. True statement, correct answer. If it comes out false, you almost certainly missed a flip.

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One-Step Inequalities

Solve it like an equation

The sign-flip rule

Graph the solution

Worked examples

Example 1: subtraction, no flip

Example 2: division by a positive, no flip

Example 3: division by a negative — flip

Try one yourself

Common questions

Keep learning

Introduction to Inequalities

Multi-Step Inequalities

Inequalities with Variables on Both Sides

One-Step Equations (Add / Subtract)