Solving Linear Inequalities: Why the Sign Flips with Negatives

Q: What does an open circle mean on a number line?

An open circle means the endpoint is not included, as with . A filled circle means it is included, as with ≤ and ≥.

What $<$ , $\le$ , $>$ and $\ge$ mean, how to show solutions on a number line with open and closed circles, how to solve an inequality, and exactly why dividing by a negative flips the sign.

4 min read Supports 3 lessons

An equation like $x = 2$ has one answer. An inequality like $x > 2$ has infinitely many: $3$ , $2.5$ , $2.001$ , and every other number bigger than $2$ .

x < 2 \;\;\text{less than} \qquad x \le 2 \;\;\text{at most} \qquad x > 2 \;\;\text{greater than} \qquad x \ge 2 \;\;\text{at least}

On the number line

Because the answer is a range, it is best shown as a shaded stretch of the number line. An open circle means the endpoint is not included ( $<$ or $>$ ); a filled circle means it is ( $\le$ or $\ge$ ).

$x > 2$ : an open circle at 2, shading to the right.

$x \le -1$ : a filled circle at $-1$ , shading to the left.

Solving: almost like an equation

You can add or subtract anything on both sides, and multiply or divide both sides by a positive number, without changing the inequality. That handles most problems.

Worked example

Solve $2x + 3 < 11$ .

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Subtract $3$ .
$2x < 8$

The flip rule

-2x < 6 \;\;\Rightarrow\;\; x > -3

Multiplying or dividing both sides by a negative number reverses the inequality.

Why? Start with something true: $2 < 5$ . Multiply both sides by $-1$ and you get $-2$ and $-5$ . But $-2$ is to the right of $-5$ on the number line, so now $-2 > -5$ . Negating mirrors the number line, and a mirror reverses order.

Reflecting $2 < 5$ across zero gives $-2 > -5$ : the order flips.

$4 - 3x \ge 13$
Solve for $x$ .

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Tip

To avoid the flip altogether, move the $x$ term to the side where it will be positive: $4 - 3x \ge 13$ becomes $-9 \ge 3x$ , so $-3 \ge x$ . Same answer.

Watch out

Swapping the two sides of an inequality also reverses the sign: $5 > x$ means the same as $x < 5$ .

Checking

Test one number inside your answer and one outside. The inside value should make the original inequality true; the outside value should make it false. If both pass or both fail, look for a missed flip.

Common questions

Why does the inequality sign flip when you multiply by a negative?: Multiplying by a negative reflects numbers across zero, which reverses their order. $2 < 5$ but $-2 > -5$ .
What does an open circle mean on a number line?: An open circle means the endpoint is not included, as with $<$ and $>$ . A filled circle means it is included, as with $\le$ and $\ge$ .

Practise inequalities

The Linear Inequalities lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Why does the inequality sign flip when you multiply by a negative?

Multiplying by a negative reflects numbers across zero, which reverses their order. $2 < 5$ but $-2 > -5$ .

What does an open circle mean on a number line?

An open circle means the endpoint is not included, as with $<$ and $>$ . A filled circle means it is included, as with $\le$ and $\ge$ .