Math Core

An equation like $x = 4$ has one answer. But plenty of real rules aren't about one exact number: "you must be at least 13 to sign up," "the bag can hold at most 50 pounds." An inequality describes a whole range of numbers at once, and a number line is the clearest way to see that range.

Reading the symbols

There are four inequality symbols. Each one compares the two sides.

Symbol	Read it as	Example	Numbers that work
$<$	is less than	$x < 3$	$2$ , $0$ , $-10$ , $2.9$
$>$	is greater than	$x > 3$	$4$ , $3.1$ , $100$
$\le$	is less than or equal to	$x \le 3$	$3$ , $2$ , $-5$
$\ge$	is greater than or equal to	$x \ge 3$	$3$ , $3.5$ , $20$

A quick memory trick: the symbol always points at the smaller side. In $2 < 7$ , the narrow tip points at $2$ .

Definition

Solution of an inequality

A solution of an inequality is any number that makes it true when you substitute it for the variable. Most inequalities have infinitely many solutions, including fractions and decimals, not just whole numbers.

To test whether a number is a solution, plug it in. Is $-1$ a solution of $x \ge -3$ ? Yes, because $-1 \ge -3$ is true: $-1$ sits to the right of $-3$ on the number line. Is $-4$ a solution? No, because $-4$ is to the left of $-3$ .

Graphing on a number line

Every graph of a one-variable inequality has two parts.

The endpoint. This is the boundary number. Draw an open circle if the boundary is not a solution ( $<$ or $>$ ). Draw a closed dot if it is a solution ( $\le$ or $\ge$ ).
The shading. Shade toward all the other solutions. For "greater than," that's to the right. For "less than," that's to the left. The shading goes on forever, past the edge of the picture.

Open or closed, left or right

$<$ or $>$ : open circle (the endpoint is left out).
$\le$ or $\ge$ : closed dot (the endpoint is included).
With the variable on the left, $x > a$ or $x \ge a$ shades right, and $x < a$ or $x \le a$ shades left.

Worked example: Graphing two inequalities

Graph $x > -2$ and $x \le 4$ .

For $x > -2$ : the symbol is $>$ , so $-2$ is not included. Draw an open circle at $-2$ and shade to the right, toward the bigger numbers.

x > -2

For $x \le 4$ : the symbol includes "or equal to," so $4$ is a solution. Draw a closed dot at $4$ and shade to the left.

x ≤ 4

Check: $0$ is shaded on the first graph, and $0 > -2$ is true. $5$ is not shaded on the second, and $5 \le 4$ is false. ✓

When the variable is on the right

Sometimes an inequality is written like $6 > x$ . Read it carefully: "6 is greater than $x$ " means the same thing as " $x$ is less than 6." So $6 > x$ is the same as $x < 6$ .

To rewrite it, swap the two sides and turn the symbol around so it still points at the same thing. Then graph it the usual way.

Worked example: Reading it backward

Graph $-1 \le x$ .

Swap sides and turn the symbol: $-1 \le x$ becomes $x \ge -1$ . Closed dot at $-1$ , shaded to the right.

x ≥ -1

Check: try $2$ . The original says $-1 \le 2$ , which is true, and $2$ is in the shaded part. ✓

Common mistake

When the variable is on the right, "greater than" does not mean shade right. In $6 > x$ you see a $>$ , so it's tempting to shade toward bigger numbers. But the sentence says $x$ is the smaller side, so the solutions are the numbers less than 6, shaded to the left. Rewrite with the variable first ( $x < 6$ ), then shade.

Going from a graph to an inequality

Work backward: find the endpoint, check whether it's open or closed, then look at which way the shading goes.

Worked example: Writing the inequality

Write the inequality shown by this graph.

The endpoint is $-3$ . The circle is open, so $-3$ is not included: use $<$ or $>$ . The shading goes left, toward smaller numbers. The inequality is $x < -3$ .

Words that mean inequalities

Real situations use phrases instead of symbols.

Phrase	Symbol	Example
at least, no less than, a minimum of	$\ge$	You need at least 60 points: $p \ge 60$
at most, no more than, a maximum of	$\le$	At most 8 people fit: $n \le 8$
more than, over, above	$>$	Temperatures above 90: $t > 90$
fewer than, less than, under, below	$<$	Kids under 5 ride free: $a < 5$

Tip

"At least" and "at most" are the tricky ones. They include the number. If you need at least 60 points, then exactly 60 points is enough, so use $\ge$ with a closed dot.

Practice

Practice 1

Which number is a solution of $x \ge -2$ ?

Practice 2

Write the inequality shown by the graph.

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

Practice 3

Write the inequality shown by the graph.

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

Practice 4

Which description matches the graph of $-4 \ge x$ ?

Practice 5

What is the greatest integer that is a solution of $x < 5.5$ ?

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

Practice 6

What is the smallest integer that is a solution of $x > -3$ ?

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

Practice 7

To ride a roller coaster, you must be at least 48 inches tall. Let $h$ be a rider's height in inches. Write an inequality for the heights that are allowed.

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

Practice 8

A soccer team can have no more than 12 players on its roster. Let $p$ be the number of players. Write an inequality.

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