Lesson 5: Inequalities

Property

An inequality is used in algebra to compare two quantities that may have different values. We use four main inequality symbols:
$a < b$ is read $a$ is less than $b$
$a > b$ is read $a$ is greater than $b$
$a \leq b$ is read $a$ is less than or equal to $b$
$a \geq b$ is read $a$ is greater than or equal to $b$

Examples

Property

A statement that uses one of the symbols $>$ or $<$ is called an inequality. An inequality that uses the symbol for less than, $<$, or greater than, $>$, is called a strict inequality. A nonstrict inequality uses one of the following symbols: $\geq$ means "greater than or equal to"; $\leq$ means "less than or equal to".

Examples

• The inequality $x > 5$ represents all numbers strictly greater than 5. On a number line, this is shown with an open circle at 5 and an arrow pointing to the right.
• The inequality $y \leq -2$ represents -2 and all numbers less than it. On a number line, this is shown with a solid dot at -2 and an arrow pointing to the left.
• The values 8, 9.5, and 200 all satisfy the inequality $x \geq 8$, but 7.9 does not.

Explanation

Inequalities describe a range of possible values, not just a single answer. A strict inequality ($<$ or $>$) uses an open circle on a number line, while a non-strict one ($\leq$ or $\geq$) uses a solid dot to show the endpoint is included.

Property

Any number greater than 3 is a solution to the inequality $x > 3$.
We show the solutions to the inequality $x > 3$ on the number line by shading in all the numbers to the right of 3.
The open parentheses symbol, (, shows that the endpoint of the inequality is not included.
The open bracket symbol, [, shows that the endpoint is included.
Because the number 3 itself is not a solution, we put an open parenthesis at 3.
The graph of the inequality $x \geq 3$ is very much like the graph of $x > 3$, but now we need to show that 3 is a solution, too.
We do that by putting a bracket at $x = 3$.
We can also represent inequalities using interval notation.
The inequality $x > 3$ is expressed as $(3, \infty)$.
The symbol $\infty$ is read as 'infinity'.
The inequality $x \leq 1$ is written in interval notation as $(-\infty, 1]$.
The symbol $-\infty$ is read as 'negative infinity'.

Examples

• The inequality $x < 4$ includes all numbers to the left of 4. On a number line, we place a parenthesis at 4 and shade to the left. In interval notation, this is $(-\infty, 4)$.

• The inequality $y \geq -2$ includes -2 and all numbers to its right. On a number line, we place a bracket at -2 and shade to the right. In interval notation, this is $[-2, \infty)$.