How does an inequality differ from an equation?

An equation asks for which number both sides are equal; an inequality asks for which numbers one side is greater or smaller than the other. That is why the solution of an inequality is usually a whole set of numbers, such as every x greater than 3, rather than a single number.

Why does multiplying by a negative number flip the inequality sign?

Because multiplying by a negative number reflects the numbers about zero on the number line and reverses their order. It is true that 2 is less than 5, but after multiplying both sides by −1 we get −2 and −5, and −2 is greater than −5. For the statement to stay true, the sign has to flip.

When does the inequality sign stay unchanged?

When you add or subtract any number on both sides, and when you multiply or divide both sides by a positive number. The sign flips only when you multiply or divide by a negative number.

How do you mark the solution set on a number line?

Mark the boundary number and shade the ray on the side where the solutions lie. For a strict inequality ( ) the boundary is not part of the set — draw an open circle; for a non-strict one (≤ or ≥) it is — draw a filled circle.

What does x belongs to the interval (3, ∞) mean?

It is the same set as x > 3: every number greater than 3. The round bracket next to the 3 means that 3 itself is not in the set; for the non-strict x ≥ 3 we would write a square bracket, that is a closed end.

Can an inequality have no solutions?

Yes. If the unknowns cancel during the rearrangement and a false statement is left, such as 0 > 5, the inequality has no solutions. If a true statement is left, such as 0 < 5, every number satisfies it.

How do you check the solution of an inequality?

Take any number from the set you found and substitute it into the original inequality — it should give a true statement. Then take a number outside the set; that one should give a false statement. Two tries are enough to catch a flipped sign.

Advanced level

Linear inequalities

Q: How do you check the solution of an inequality?

Take any number from the set you found and substitute it into the original inequality — it should give a true statement. Then take a number outside the set; that one should give a false statement. Two tries are enough to catch a flipped sign.

A linear inequality has not one solution but a whole set of them. Learn the operations that keep the inequality sign as it is, memorise the single one that flips it, and learn to mark the solution set on a number line.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Linear equationsA linear equation is an equality with the unknown in the first power. Learn the operations that keep equations equivalent, solve one step by step, check the result, rearrange a formula to make another quantity its subject, and handle an unknown in a denominator.

Where this is used

Real situations where you count exactly the way this lesson teaches:

Online shopping
The shop ships free above 199, your basket stands at 164.50 and a pack of coffee costs 12.90. From 164.50 + 12.90n ≥ 199 you get n ≥ 2.68 — and since nobody sells 2.68 packs, you add three and finish at 203.20.
Monthly budget
You have 1,800 in the account, 30 days until payday and want 300 left as a buffer: 1800 − 30s ≥ 300, so −30s ≥ −1500, and dividing by a negative number gives s ≤ 50 a day. Forget to flip the sign and you read it as "at least 50 a day" — the exact opposite advice.
Courier work
A van rated at 3,500 kg gross weighs 2,150 kg empty and the driver adds 85 kg, leaving 1,265 kg for cargo. With 28 kg parcels, 28n ≤ 1265 gives n ≤ 45.2 — here the answer rounds down, to 45 parcels, because parcel forty-six is a fine.
Convert it here: Mass

All formulas

A linear inequality
$ax + b > c \quad (a \neq 0)$
one of <, >, ≤, ≥ in place of the equals sign
Adding a number to both sides
$a > b \iff a + c > b + c$
the inequality sign stays as it is
Multiplying by a positive number
$a > b \iff ac > bc \quad (c > 0)$
the inequality sign stays as it is
Multiplying by a negative number
$a > b \iff ac < bc \quad (c < 0)$
the inequality sign FLIPS
Interval notation
$x > 3 \iff x \in (3, \infty)$
a round bracket — the endpoint is not included
A non-strict inequality
$x \ge 3 \iff x \in [3, \infty)$
a square bracket — the endpoint is included

A linear inequality looks like a linear equation, except that instead of an equals sign it carries one of $<$ , $>$ , $\le$ , $\ge$ :

ax + b > c \quad (a \neq 0)

The difference in the answer is fundamental. A linear equation usually has one solution; an inequality has a whole set of them. The answer is not a number but a condition such as $x > 3$ .

What you may do to an inequality

Inequalities are rearranged just like equations, with one exception covered below. The following keep the inequality sign unchanged:

a > b \iff a + c > b + c

a > b \iff a \cdot c > b \cdot c \quad (c > 0)

So terms may be moved to the other side (changing sign, as in equations), and both sides may be divided by a positive number.

The one exception: multiplying and dividing by a negative number

When both sides are multiplied or divided by a negative number, the inequality sign must be flipped:

a > b \iff a \cdot c < b \cdot c \quad (c < 0)

Why? Because multiplying by a negative number reflects the number line about zero and reverses the order of numbers. $2 < 5$ is true, but after multiplying both sides by $-1$ we have $-2$ and $-5$ , and $-2 > -5$ . If the sign stayed as it was, a true statement would turn false.

Solve: −3x + 1 < 10.

Note that the flip is caused by multiplying or dividing by a negative number, not by a minus appearing somewhere in the problem. Moving $-3$ to the other side changes nothing about the sign.

The solution set on a number line

The solution of an inequality is easiest to see on a number line: mark the boundary number and take the whole ray on the side where the solutions lie.

The solutions of x > 3 are all the numbers to the right of 3; the circle at 3 is open, because 3 itself is not one of them.

Whether the boundary belongs to the set is decided by the kind of inequality — and that is exactly what the circle at its end says:

strict ( $<$ , $>$ ) — the boundary is not included; an open circle, as in the drawing above,
non-strict ( $\le$ , $\ge$ ) — the boundary is included; a filled circle.

The same set is written more briefly as an interval:

x > 3 \iff x \in (3, \infty) \qquad x \ge 3 \iff x \in [3, \infty)

A round bracket marks an end not in the set, a square bracket one that is. Infinity always takes a round bracket, because infinity is not a number.

Solve and write as an interval: 4x − 5 ≥ 2x + 7.

Inequalities in practice

Inequalities answer questions of the "at most" and "at least" kind. A delivery costs 40 plus 6 per parcel, and the budget is 100. How many parcels fit the budget?

40 + 6x \le 100 \quad \big| -40

6x \le 60 \quad \big| : 6

x \le 10

The solution set is every $x \le 10$ , but a number of parcels is a non-negative whole number — so the answer is: at most ten. That is typical of word problems: mathematics gives the solution set, and the meaning of the problem narrows it down.

In the exercises below, enter the whole solution set, for example x > 3 or x ≥ 3 (x >= 3 works too). The boundary number alone is not enough — the direction counts as well.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

4x + 1 > 21

Common mistakes

Not flipping the sign after dividing by a negative number — the single most common mistake here.
Flipping the sign when moving a term across — moving is addition, so the sign stays.
Mixing up strict and non-strict — $x > 3$ excludes 3, $x \ge 3$ includes it.
Answering with the boundary alone — $3$ is not the answer; $x > 3$ is.
Ignoring what the problem is about — a number of parcels or people is never a fraction or negative.

Formula card

Topic: Linear inequalities

A linear inequality
$ax + b > c \quad (a \neq 0)$
one of <, >, ≤, ≥ in place of the equals sign
Adding a number to both sides
$a > b \iff a + c > b + c$
the inequality sign stays as it is
Multiplying by a positive number
$a > b \iff ac > bc \quad (c > 0)$
the inequality sign stays as it is
Multiplying by a negative number
$a > b \iff ac < bc \quad (c < 0)$
the inequality sign FLIPS
Interval notation
$x > 3 \iff x \in (3, \infty)$
a round bracket — the endpoint is not included
A non-strict inequality
$x \ge 3 \iff x \in [3, \infty)$
a square bracket — the endpoint is included

The solution set of x > 3: every number to the right of 3, but not 3 itself.

Frequently asked questions

A balance scale in perfect equilibrium rendered as a geometric form, with a straight line crossing an axis like an orbital path, against a cosmic nebula in violet and magenta

Algebra

What x really means — unknown, variable and parameter

The letter x does not always mean the same thing. Sometimes it is a number you are hunting for, sometimes a quantity that varies continuously, sometimes a blank waiting for any number at all. Here is where algebraic notation came from, why it took two thousand years to arrive, what exactly separates an unknown from a variable and a parameter, and why an inequality flips when you multiply by a negative number.

11 min