Math Core

An equation like $x + 3 = 10$ has exactly one answer. Many real questions don't: you must be at least 16 to get a driver's permit, an elevator holds no more than 1,500 pounds, a phone plan lets you use up to 5 GB. Each of these describes a whole range of numbers, and the tool for writing a range is an inequality.

The inequality symbols

Definition

Inequality

An inequality is a statement that compares two expressions using one of these symbols:

symbol	read as	example
$<$	is less than	$x < 4$
$>$	is greater than	$x > -1$
$\le$	is less than or equal to	$x \le 10$
$\ge$	is greater than or equal to	$x \ge 0$

The symbol always opens toward the larger side, like a mouth that wants the bigger number. In $2 < 9$ the wide end faces $9$ .

An inequality can be read in either direction. The statement $x < 4$ (" $x$ is less than $4$ ") says exactly the same thing as $4 > x$ (" $4$ is greater than $x$ "). When you flip the sides, you flip the symbol so it still points at the same thing.

Solutions of an inequality

A solution of an inequality is any value of the variable that makes it true. To test a value, substitute it and check whether the comparison holds.

For $x + 5 \ge 8$ :

value	substitute	true?
$x = 2$	$2 + 5 = 7$ , and $7 \ge 8$	no
$x = 3$	$3 + 5 = 8$ , and $8 \ge 8$	yes
$x = 3.5$	$3.5 + 5 = 8.5$ , and $8.5 \ge 8$	yes
$x = 100$	$105 \ge 8$	yes

Every number $3$ or greater works, including fractions and decimals like $3.5$ . So this inequality has infinitely many solutions. You can't list them all, but you can describe them with a simpler inequality, $x \ge 3$ , or draw them.

Graphing on a number line

The graph of an inequality in one variable shades every solution on a number line. Two choices decide what the graph looks like.

The circle at the boundary. Use a closed (filled) circle when the boundary number is a solution ( $\le$ or $\ge$ ). Use an open (hollow) circle when it is not ( $<$ or $>$ ).
The direction of shading. Shade toward the numbers that make the statement true. When the variable is on the left, "less than" shades to the left and "greater than" shades to the right.

Here is $x \ge 3$ . The filled circle shows that $3$ is included, and the arrow shows that the solutions continue forever to the right.

x ≥ 3

And here is $x < -1$ . The open circle shows that $-1$ itself is not a solution, but every number just below it, like $-1.001$ , is.

x < -1

Reading and drawing graphs

inequality	circle	shading
$x > a$	open at $a$	to the right
$x \ge a$	closed at $a$	to the right
$x < a$	open at $a$	to the left
$x \le a$	closed at $a$	to the left

Common mistake

When the variable is on the right, as in $6 > x$ , don't shade in the direction the symbol "points." Rewrite it with the variable first: $6 > x$ means $x < 6$ , so the graph shades left from an open circle at $6$ . Test a number if you're unsure: $0$ makes $6 > 0$ true, so the side containing $0$ gets shaded.

Worked example: Graphing an inequality

Graph $y \le 2.5$ .

The symbol $\le$ includes the boundary, so draw a closed circle at $2.5$ . "Less than" means shade to the left.

y ≤ 2.5

Check with a test point: $y = 0$ gives $0 \le 2.5$ , which is true, and $0$ is on the shaded side.

Translating words into inequalities

Everyday phrases map onto the four symbols. The trickiest ones are "at least" and "at most," because they use no word like "greater" or "less."

phrase	symbol
is more than, exceeds, is above	$>$
is less than, is below, fewer than	$<$
at least, no less than, a minimum of	$\ge$
at most, no more than, a maximum of, up to	$\le$

A quick way to remember: "at least $16$ " means $16$ is the least you can have, so $16$ and everything above it is allowed.

Worked example: Words to symbols

Write an inequality for each statement.

A roller coaster rider's height $h$ must be at least $48$ inches.
The number of people $p$ in the boat can be no more than $6$ .
The temperature $t$ stayed above $-4$ degrees all night.
You have $30. The total cost $c$ of your meal must not exceed that.

Solutions.

"At least" includes $48$ : $h \ge 48$ .
"No more than" includes $6$ : $p \le 6$ .
"Above" does not include $-4$ : $t > -4$ .
"Must not exceed" means the cost can equal $30 but not go over: $c \le 30$ .

Some situations also have a hidden restriction. In part 2 above, $p$ counts people, so it must be a whole number. The solutions are really $0, 1, 2, \dots, 6$ , even though the inequality $p \le 6$ on its own allows $5.5$ and $-3$ . Always ask whether the answer makes sense for the situation.

Writing an inequality from a graph

To go backward from a graph, read the three pieces of information it shows: the boundary number, the type of circle, and the direction of shading.

Worked example: Reading a graph

Write the inequality shown by each graph.

Solutions.

The boundary is $-2$ with an open circle, so $-2$ is not included. The shading goes right: $x > -2$ .
The boundary is $7$ with a closed circle, so $7$ is included. The shading goes left: $x \le 7$ .

Tip

To check any graph, pick one number in the shaded part and one number outside it. The shaded number should make the inequality true and the other should make it false.

Practice

Practice 1

Which value is a solution of $2x - 1 > 7$ ?

Practice 2

Write an inequality: your speed $s$ may be at most $65$ miles per hour.

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

Practice 3

Write an inequality: a club needs at least $12$ members $n$ to be recognized by the school.

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

Practice 4

Write the inequality shown by the graph.

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

Practice 5

Write the inequality shown by the graph.

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

Practice 6

Which description matches the graph of $-5 < x$ ?

Practice 7

A carry-on bag can weigh no more than $22$ pounds. Which inequality describes the allowed weights $w$ , and which weight is not allowed?

Practice 8

How many integers are solutions of both $x > -2$ and $x \le 3$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.