Math Core

Not every rule has just one right number. "You must be at least 48 inches tall to ride" is true for 48 inches, 50 inches, 61.5 inches, and many more. To describe a whole range of numbers like that, you use an inequality.

Inequality symbols

An inequality compares two amounts that might not be equal. There are four symbols you'll use.

symbol	meaning	example
$>$	is greater than	$x > 4$
$<$	is less than	$x < 4$
$\ge$	is greater than or equal to	$x \ge 4$
$\le$	is less than or equal to	$x \le 4$

The small line under $\ge$ and $\le$ means "or equal to." So $x \ge 4$ includes $4$ itself, but $x > 4$ does not.

Definition

Solution of an inequality

A solution of an inequality is any value of the variable that makes the inequality true. Most inequalities have infinitely many solutions.

For $x > 4$ , the numbers $5$ , $4.1$ , $10$ and $1{,}000$ are all solutions. The number $4$ is not, because $4 > 4$ is false. Neither is $3$ .

Writing inequalities from words

Everyday phrases tell you which symbol to use.

phrase	symbol
more than, greater than, above, over	$>$
less than, fewer than, below, under	$<$
at least, no less than, a minimum of	$\ge$
at most, no more than, a maximum of	$\le$

Worked example: From words to symbols

Write an inequality for each sentence.

You must be at least 48 inches tall. Use $h$ for height.
The elevator can carry at most 1,500 pounds. Use $w$ for weight.
The temperature stayed below $-2$ degrees. Use $t$ for temperature.

Solutions.

"At least 48" means 48 or more: $h \ge 48$ .
"At most 1,500" means 1,500 or less: $w \le 1500$ .
"Below $-2$ " means less than $-2$ : $t < -2$ .

Common mistake

"At least" and "at most" sound backward to many people. "At least 10" means $10$ is the smallest allowed value, so use $\ge$ . "At most 10" means $10$ is the largest allowed value, so use $\le$ . Test a number: if you need at least 10 points, 12 points is fine and 8 is not.

Graphing on a number line

You can't list infinitely many solutions, but you can draw them. A graph of an inequality on a number line has two parts:

A circle at the boundary number. An open circle means the number is not included ( $>$ or $<$ ). A closed (filled) circle means it is included ( $\ge$ or $\le$ ).
A shaded ray showing all the other solutions. Shade to the right for "greater," and to the left for "less."

Here is $x > 2$ . The circle at $2$ is open, and the shading goes right.

x > 2

Here is $x \le -1$ . The circle at $-1$ is filled, and the shading goes left.

x ≤ -1

Graphing an inequality

Find the boundary number and put a circle there: open for $>$ or $<$ , closed for $\ge$ or $\le$ .
Shade toward the solutions: right for greater than, left for less than.

Worked example: Graph a real-world inequality

A car ride holds no more than 5 people. Write and graph an inequality for the number of people, $p$ .

"No more than 5" means 5 or fewer, so $p \le 5$ . Put a closed circle at $5$ and shade to the left.

p ≤ 5

In real life, $p$ must be a whole number and can't be negative, so the people could only be $0, 1, 2, 3, 4$ or $5$ . The inequality still describes the rule.

Worked example: Read a graph

Write the inequality shown by this graph.

The circle at $-3$ is open, so $-3$ is not included. The shading goes right, toward greater numbers. The inequality is $x > -3$ .

Tip

Check your graph by testing one shaded number. For $x > -3$ , pick $0$ : is $0 > -3$ ? Yes, so $0$ should be shaded, and it is.

Practice

Practice 1

Write an inequality: a number $x$ is greater than $4$ .

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Practice 2

To see a movie alone, you must be at least 13 years old. Write an inequality for the allowed ages, $a$ . (Type $\ge$ as >=.)

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Practice 3

Which number is a solution of $x < -3$ ?

Practice 4

Write the inequality shown by the graph. Use $x$ as the variable. (Type $\le$ as <=.)

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Practice 5

A small bridge can hold no more than 1,200 pounds. Write an inequality for the weight $w$ it can safely hold.

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Practice 6

On a number line graph of $t > -1$ , what should appear at $-1$ ?

Practice 7

List all the whole numbers that are solutions of $n < 4$ .

Separate answers with commas, e.g. 2, -5

Practice 8

Overnight, the temperature dropped below $-5$ degrees Fahrenheit. Write an inequality for the temperature $t$ .

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5