How to Solve One-Step Inequalities

Quick answer

Solve a one-step inequality exactly like a one-step equation, by doing the same inverse operation to both sides. One rule is different: if you multiply or divide both sides by a negative number, you must flip the inequality sign. So -2x < 6 becomes x > -3, not x < -3.

What you'll learn

Solve one-step inequalities using inverse operations
Explain why the inequality sign flips when multiplying or dividing by a negative
Graph a solution set on a number line

What an inequality is

An equation says two things are equal. An inequality says one is bigger or smaller:

x + 3 < 10

The four symbols:

Symbol	Means	Example
$<$	less than	$2 < 5$
$>$	greater than	$7 > 4$
$\le$	less than or equal to	$x \le 3$ allows $x = 3$
$\ge$	greater than or equal to	$x \ge 3$ allows $x = 3$

The big difference from an equation: an equation usually has one answer, but an inequality has many. Solving $x + 3 = 10$ gives $x = 7$ . Solving $x + 3 < 10$ gives $x < 7$ — and $6$ , $0$ , $-45$ and $6.999$ all work.

Definition

The solution set is every number that makes the inequality true. For $x < 7$ that is every number to the left of $7$ on the number line.

Solving works exactly like an equation

Do the same inverse operation to both sides, exactly as you already know:

x + 3 < 10

x + 3 - 3 < 10 - 3

x < 7

Adding and subtracting behave normally, and so does multiplying or dividing by a positive number. If that were the whole story, this lesson would be over. It is not.

Why the sign flips for a negative

This is the one rule that catches everyone, and it is worth understanding rather than memorising, because a memorised rule gets applied at the wrong moment.

Start with something certainly true:

2 < 5

Now multiply both sides by $-1$ :

-2 \quad ? \quad -5

Is $-2$ less than $-5$ ? No. On the number line $-2$ sits to the right of $-5$ , so $-2$ is greater. The correct statement is:

-2 > -5

The numbers kept their sizes but swapped their order. That is what multiplying by a negative does: it reflects the whole number line about zero, and a reflection reverses left and right.

The inequality symbol is a claim about order. If the order reverses, the symbol has to reverse with it, or the sentence stops being true.

Multiply or divide both sides by a negative → flip the sign. Add or subtract anything → do not flip.

Adding does not flip because it slides the number line instead of reflecting it. Slide two points the same distance and the one on the left is still on the left.

Graphing the solution

A picture of the solution set uses a circle at the boundary and an arrow for the direction:

Symbol	Circle	Arrow points
$x < 7$	open ○ at 7	left
$x \le 7$	closed ● at 7	left
$x > 7$	open ○ at 7	right
$x \ge 7$	closed ● at 7	right

Open means “not included”, closed means “included”. That is the whole convention.

Worked examples

Common mistakes

Practice problems

Solve $x + 8 < 15$ .

Hint

Subtract from both sides. Nothing is multiplied, so nothing flips.

Answer

$x < 7$

Full solution

Subtract $8$ from both sides: $x < 7$ .

Check: $x = 6$ gives $14 < 15$ ✓; $x = 7$ gives $15 < 15$ , false, so $7$ is correctly excluded.
Solve $y - 3 \ge 9$ .

Answer

$y \ge 12$

Full solution

Add $3$ to both sides: $y \ge 12$ .

Check: $y = 12$ gives $9 \ge 9$ ✓.
Solve $6k \le 42$ .

Answer

$k \le 7$

Full solution

Divide both sides by $6$ . It is positive, so the sign stays: $k \le 7$ .

Check: $k = 7$ gives $42 \le 42$ ✓.
Solve $-3x > 12$ .

Hint

You are dividing by a negative. What happens to the sign?

Answer

$x < -4$

Full solution

Divide both sides by $-3$ and flip: $x < -4$ .

Check: $x = -5$ gives $-3(-5) = 15 > 12$ ✓, and $-5 < -4$ ✓.
Solve $\tfrac{m}{-2} \le 5$ .

Answer

$m \ge -10$

Full solution

Multiply both sides by $-2$ and flip: $m \ge -10$ .

Check: $m = -10$ gives $\tfrac{-10}{-2} = 5 \le 5$ ✓.
Solve $-x < 4$ .

Hint

$-x$ means $-1 \cdot x$ . Divide by $-1$ .

Answer

$x > -4$

Full solution

Divide both sides by $-1$ and flip: $x > -4$ .

Check: $x = 0$ gives $0 < 4$ ✓, and $0 > -4$ ✓.
Is $x = -2$ a solution of $-4x \ge 8$ ?

Answer

Yes.

Full solution

Substitute directly: $-4(-2) = 8$ , and $8 \ge 8$ is true, so $-2$ is a solution.

Solving confirms it: divide by $-4$ and flip to get $x \le -2$ , which includes $-2$ .
A ride at a fair requires you to be at least 48 inches tall. Write an inequality for the allowed heights $h$ , and say whether someone 48 inches tall can ride.

Hint

“At least” means the boundary value is allowed.

Answer

$h \ge 48$ , and yes, they can ride.

Full solution

“At least 48 inches” means 48 counts, so the symbol includes equality: $h \ge 48$ .

On a number line this is a closed circle at $48$ with an arrow to the right. Someone exactly 48 inches tall satisfies $48 \ge 48$ , so they can ride.

Frequently asked questions

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

Multiplying by a negative reverses the order of every number on the number line. 2 is less than 5, but -2 is greater than -5. Since the order reverses, the symbol describing that order has to reverse with it, or the statement becomes false.

Does the sign flip when you add or subtract a negative?

No. Adding or subtracting shifts both sides the same distance along the number line, so their order does not change. Only multiplying or dividing by a negative reverses order, and only then does the sign flip.

Why is the circle sometimes open and sometimes closed?

An open circle means the endpoint is not included, which matches < and >. A closed circle means it is included, which matches ≤ and ≥. The circle is a picture of whether the boundary number itself is a solution.

What to learn next

How to Solve Multi-Step InequalitiesSolve multi-step inequalities by distributing, combining like terms, and undoing operations in reverse order, including when to flip the inequality sign.

Key terms in this lesson

Solution set: The solution set is every value that makes an equation or inequality true. An equation often has one solution, an inequality usually has infinitely many, and some have none at all.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.7.EE.B.4bExpressions and EquationsSolve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

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Quick answer

What you'll learn

What an inequality is

Solving works exactly like an equation

Why the sign flips for a negative

Graphing the solution

Worked examples

Common mistakes

Practice problems

Frequently asked questions

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

Does the sign flip when you add or subtract a negative?

Why is the circle sometimes open and sometimes closed?

What to learn next

Key terms in this lesson

Standards alignment

More Algebra 1 lessons