6.1: Inequality Expressions
What if the maximum occupancy of an elevator were listed at 20 people? How could you graph the number of allowable occupants on the elevator? After completing this Concept, you'll be able to write and graph inequalities like this one.
Watch This
CK12 Foundation: 0601S Graphing Inequalities (H264)
Guidance
Dita has a budget of $350 to spend on a rental car for an upcoming trip, but she wants to spend as little of that money as possible. If the trip will last five days, what range of daily rental rates should she be willing to consider?
Like equations, inequalities show a relationship between two expressions. We solve and graph inequalities in a similar way to equations—but when we solve an inequality, the answer is usually a set of values instead of just one value.
When writing inequalities we use the following symbols:
> is greater than
\begin{align*}\ge\end{align*}
< is less than
\begin{align*}\le\end{align*}
Write and Graph Inequalities in One Variable on a Number Line
Let’s start with the simple inequality \begin{align*}x > 3\end{align*}
We read this inequality as “\begin{align*}x\end{align*}
Consider another simple inequality: \begin{align*}x \le 4\end{align*}
We read this inequality as “\begin{align*}x\end{align*}
Notice that we use an empty circle for the endpoint of a strict inequality (like \begin{align*}x > 3\end{align*}
Example A
Graph the following inequalities on the number line.
a) \begin{align*}x< 3\end{align*}
b) \begin{align*}x \ge 6\end{align*}
c) \begin{align*}x > 0 \end{align*}
Solution
a) The inequality \begin{align*}x < 3\end{align*}
b) The inequality \begin{align*}x \ge 6\end{align*}
c) The inequality \begin{align*}x > 0\end{align*}
Example B
Write the inequality that is represented by each graph.
a)
b)
c)
Solution
a) \begin{align*}x \le 12\end{align*}
b) \begin{align*}x >540\end{align*}
c) \begin{align*}x < 6.5\end{align*}
Inequalities appear everywhere in real life. Here are some simple examples of realworld applications.
Example C
Write each statement as an inequality and graph it on the number line.
a) You must maintain a balance of at least $2500 in your checking account to get free checking.
b) You must be at least 48 inches tall to ride the “Thunderbolt” Rollercoaster.
c) You must be younger than 3 years old to get free admission at the San Diego Zoo.
Solution
a) The words “at least” imply that the value of $2500 is included in the solution set, so the inequality is written as \begin{align*}x \ge 2500\end{align*}
b) The words “at least” imply that the value of 48 inches is included in the solution set, so the inequality is written as \begin{align*}x \ge 48\end{align*}
c) The inequality is written as \begin{align*}x < 3\end{align*}
Watch this video for help with the Examples above.
CK12 Foundation: Graphing Inequalities
Vocabulary
 The answer to an inequality is usually an interval of values.
Guided Practice
1. Graph the inequality \begin{align*}x \le 8\end{align*}
2. Write the inequality that is represented by the graph below.
3. Write the statement, "the speed limit on the interstate is 65 miles per hour or less" as an inequality .
Solution
1. The inequality \begin{align*}x \le 8\end{align*}
2. \begin{align*}x \ge 85\end{align*}
3. Speed limit means the highest allowable speed, so the inequality is written as \begin{align*}x \le 65\end{align*}
Practice
 Write the inequality represented by the graph.
 Write the inequality represented by the graph.
 Write the inequality represented by the graph.
 Write the inequality represented by the graph.
Graph each inequality on the number line.

\begin{align*}x < 35\end{align*}
x<−35 
\begin{align*}x > 17\end{align*}
x>−17 
\begin{align*}x \ge 20\end{align*}
x≥20 
\begin{align*}x \le 3\end{align*}
x≤3  \begin{align*}x \ge 5\end{align*}
 \begin{align*}x > 20\end{align*}
Image Attributions
Here you'll learn how to write and graph inequalities in one variable on a number line.