What if you were given a linear inequality like \begin{align*}|y| \ge -5\end{align*}

### Watch This

CK-12 Foundation: 0611S Graphing Linear Inequalities in the Coordinate Plane (H264)

### Guidance

A **linear inequality** in two variables takes the form \begin{align*}y > mx+b\end{align*}

When we graph a line in the coordinate plane, we can see that it divides the plane in half:

The solution to a linear inequality includes all the points in one half of the plane. We can tell which half by looking at the inequality sign:

> The solution set is the half plane above the line.

\begin{align*}\ge\end{align*}

< The solution set is the half plane below the line.

\begin{align*}\le\end{align*}

For a strict inequality, we draw a **dashed line** to show that the points in the line *are not* part of the solution. For an inequality that includes the equals sign, we draw a **solid line** to show that the points on the line *are* part of the solution.

#### Example A

This is a graph of \begin{align*}y \ge mx + b\end{align*}

This is a graph of \begin{align*}y < mx + b\end{align*}

**Graph Linear Inequalities in One Variable in the Coordinate Plane**

In the last few sections we graphed inequalities in one variable on the number line. We can also graph inequalities in one variable on the coordinate plane. We just need to remember that when we graph an equation of the type \begin{align*}x = a\end{align*}

#### Example B

*Graph the inequality \begin{align*}x > 4\end{align*} x>4 on the coordinate plane.*

**Solution**

First let’s remember what the solution to \begin{align*}x > 4\end{align*}

The solution to this inequality is the set of all real numbers \begin{align*}x\end{align*}

In two dimensions, the solution still consists of all the points to the right of \begin{align*}x = 4\end{align*}

The line \begin{align*}x = 4\end{align*}

#### Example C

Graph the inequality \begin{align*}|y| < 5\end{align*}

**Solution**

The absolute value inequality \begin{align*}|y| < 5\end{align*}

\begin{align*}y > -5 \quad \text{and} \quad y < 5\end{align*}

In other words, the solution is all the coordinate points for which the value of \begin{align*}y\end{align*}**and** smaller than 5. The solution is represented by the plane between the horizontal lines \begin{align*}y = -5\end{align*} and \begin{align*}y = 5\end{align*}.

Both horizontal lines are dashed because points on the lines are not included in the solution.

Watch this video for help with the Examples above.

CK-12 Foundation: Graphing Inequalities in the Coordinate Plane

### Guided Practice

*Graph the inequality \begin{align*}|x| \ge 2\end{align*}.*

**Solution:**

The absolute value inequality \begin{align*}|x| \ge 2\end{align*} can be re-written as a compound inequality:

\begin{align*}x \le -2 \quad \text{or} \quad x \ge 2\end{align*}

In other words, the solution is all the coordinate points for which the value of \begin{align*}x\end{align*} is smaller than or equal to -2 **or** greater than or equal to 2. The solution is represented by the plane to the left of the vertical line \begin{align*}x = -2\end{align*} and the plane to the right of line \begin{align*}x = 2\end{align*}.

Both vertical lines are solid because points on the lines are included in the solution.

### Explore More

Graph the following inequalities on the coordinate plane.

- \begin{align*}x < 20\end{align*}
- \begin{align*}y \ge -5\end{align*}
- \begin{align*}x > 0.5\end{align*}
- \begin{align*}x \le \frac{1}{2}\end{align*}
- \begin{align*}y > -\frac{2}{3}\end{align*}
- \begin{align*}y < -0.2\end{align*}
- \begin{align*}|x| > 10\end{align*}
- \begin{align*}|y| \le 7\end{align*}
- \begin{align*}|y| < \frac{1}{3}\end{align*}
- \begin{align*}|x| \ge -10\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 6.11.

### Texas Instruments Resources

*In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9616.*