Introduction to Inequalities

Q: Which way do I shade?

Toward the solutions. Read the inequality with the variable first: $x > 5$ says $x$ is bigger than $5$, and bigger numbers are to the right. If the variable is on the right, flip your reading first: $3 3$.

Q: What's the difference between $x > 4$ and $x \geq 4$?

One number: $4$ itself. In $x \geq 4$, the value $4$ is a solution and the graph uses a closed circle. In $x > 4$, it isn't, and the graph uses an open circle. Everything above $4$ is a solution either way.

An inequality compares two values and says one is larger or smaller than the other, instead of exactly equal. Where an equation like $x = 5$ has one answer, an inequality like $x > 5$ has infinitely many — $5.1$ , $6$ , $100$ , and every other number bigger than $5$ all work.

That's why inequalities get graphed on a number line: you can't list every solution, but you can shade the whole region where the solutions live. Once you know the four symbols and the two circle types, reading and graphing inequalities is fast.

The four symbols

$>$ means greater than. $<$ means less than. The open end of the symbol always faces the larger value — $7 > 3$ reads left to right as “seven is greater than three.”

$\geq$ means greater than or equal to, and $\leq$ means less than or equal to. That little line under the symbol matters: it means the boundary number itself counts as a solution. In $x \leq 7$ , the number $7$ works; in $x < 7$ , it does not.

Watch for word-problem phrases too. “At least $12$ ” means $x \geq 12$ — the value can be $12$ or more. “At most $10$ ” means $x \leq 10$ — it can be $10$ but no more.

Graphing on a number line

Every graph has two decisions: what kind of circle goes at the boundary number, and which direction to shade.

Circle: use an open circle for the strict symbols $>$ and $<$ , because the boundary number is not a solution. Use a closed (filled) circle for $\geq$ and $\leq$ , because it is.

Shading: shade in the direction of the solutions. Greater than ( $>$ or $\geq$ ) shades to the right, toward larger numbers. Less than ( $<$ or $\leq$ ) shades to the left, toward smaller numbers.

The graph below shows $x \geq 2$ : a closed circle at $2$ (because $\geq$ includes the boundary) with shading running right toward the larger numbers.

Checking whether a number is a solution

To test any number, substitute it in and ask whether the statement is true. Is $4$ a solution to $x > 4$ ? Substitute: $4 > 4$ is false — $4$ is not greater than itself — so no. Is $9$ a solution? $9 > 4$ is true, so yes. This ten-second check is also how you verify a graph you've drawn: pick a number in your shaded region and make sure it works.

Worked examples

Example 1: graph a strict inequality

Graph $x > 5$ on a number line.

Identify the boundary number

5

The symbol

>

is strict, so use an open circle at

5

Solutions are greater than

5

, so shade to the right

Answer: Open circle at $5$ , shaded to the right

Example 2: graph with the boundary included

Graph $x \leq -1$ on a number line.

Identify the boundary number

-1

The symbol

\leq

includes equal, so use a closed circle at

-1

Solutions are less than

-1

, so shade to the left

Answer: Closed circle at $-1$ , shaded to the left

Example 3: translate a phrase

Write an inequality for “you must be at least $13$ years old,” using $a$ for age.

“At least

13

” means

13

counts and anything above counts

That is greater than or equal to

a \geq 13

Answer: $a \geq 13$

Try one yourself

Common questions

How do I remember which circle to use?

Ask whether the boundary number itself is a solution. If the symbol has the “or equal to” line ( $\geq$ or $\leq$ ), the answer is yes — fill the circle in. If it's strict ( $>$ or $<$ ), the answer is no — leave the circle open.

Which way do I shade?

Toward the solutions. Read the inequality with the variable first: $x > 5$ says $x$ is bigger than $5$ , and bigger numbers are to the right. If the variable is on the right, flip your reading first: $3 < x$ means the same thing as $x > 3$ .

What's the difference between

x > 4

and

x \geq 4

One number: $4$ itself. In $x \geq 4$ , the value $4$ is a solution and the graph uses a closed circle. In $x > 4$ , it isn't, and the graph uses an open circle. Everything above $4$ is a solution either way.

Why do inequalities have infinitely many solutions?

Because they describe a region, not a single point. $x > 5$ is satisfied by $5.01$ , $5.5$ , $6$ , $1{,}000$ , and so on forever. That's exactly why we graph the solution as a shaded ray instead of listing answers.

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Introduction to Inequalities

The four symbols

Graphing on a number line

Checking whether a number is a solution

Worked examples

Example 1: graph a strict inequality

Example 2: graph with the boundary included

Example 3: translate a phrase

Try one yourself

Common questions

Keep learning

One-Step Inequalities

Multi-Step Inequalities

Graphing Linear Inequalities

Compound Inequalities