Lesson 5: Solving Compound Inequalities

Property

A compound inequality is made up of two inequalities connected by the word “and” or the word “or.”
To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement.
We solve each inequality separately and then consider the two solutions.

Examples

• The statement $x > 2$ and $x < 7$ is a compound inequality. A value like $x=5$ is a solution because it satisfies both conditions.

• The statement $y \leq 0$ or $y \geq 4$ is a compound inequality. A value like $y=6$ is a solution because it satisfies the second condition, even though it fails the first.

Property

We show all the solutions to an inequality on the number line by shading. An open parenthesis shows an endpoint is not a solution; a bracket shows an endpoint is a solution. In interval notation, we use parentheses or brackets to show the interval of solutions. The symbol $\infty$ is read as “infinity,” and $-\infty$ is read as “negative infinity.”

Examples

• The inequality $x > 5$ is written in interval notation as $(5, \infty)$. On a number line, an open parenthesis is placed at 5, and the line is shaded to the right.
• The inequality $y \leq -1$ is written in interval notation as $(-\infty, -1]$. On a number line, a bracket is placed at -1, and the line is shaded to the left.
• The inequality $-4 \leq z < 2$ is written as $[-4, 2)$. On a number line, there is a bracket at -4, a parenthesis at 2, and the line between them is shaded.

Explanation

Think of graphing an inequality as drawing a map of all its solutions. A parenthesis '(' or ')' means the endpoint is not included, while a bracket '[' or ']' means it is. Interval notation is just a shorthand for this map.

Property

A double inequality is a compound inequality such as $a < x < b$. It is equivalent to $a < x$ and $x < b$.
To solve, perform the same operation on all three parts of the inequality to isolate the variable in the middle.

$a \leq x \leq b$ is equivalent to $a \leq x$ and $x \leq b$

$a > x > b$ is equivalent to $a > x$ and $x > b$