6.11: Graphs of Inequalities in One Variable
What if you were given a linear inequality like \begin{align*}y \ge 5\end{align*}
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CK12 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*}
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*}
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.
CK12 Foundation: Graphing Inequalities in the Coordinate Plane
Vocabulary
 Inequalities of the type \begin{align*}x<a\end{align*}
x<a can be rewritten as “\begin{align*}a < x < a\end{align*}−a<x<a .”  Inequalities of the type \begin{align*}x>b\end{align*}
x>b can be rewritten as “\begin{align*}x < b\end{align*}x<−b or \begin{align*}x > b\end{align*}x>b .” 
Horizontal lines are defined by the equation \begin{align*}y=\end{align*}
y= constant and vertical lines are defined by the equation \begin{align*}x= \end{align*}x= constant.  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.
 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*}
Guided Practice
Graph the inequality \begin{align*}x \ge 2\end{align*}
Solution:
The absolute value inequality \begin{align*}x \ge 2\end{align*}
\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*}
Both vertical lines are solid because points on the lines are included in the solution.
Practice
Graph the following inequalities on the coordinate plane.

\begin{align*}x < 20\end{align*}
x<20 
\begin{align*}y \ge 5\end{align*}
y≥−5 
\begin{align*}x > 0.5\end{align*}
x>0.5 
\begin{align*}x \le \frac{1}{2}\end{align*}
x≤12 
\begin{align*}y > \frac{2}{3}\end{align*}
y>−23 
\begin{align*}y < 0.2\end{align*}
y<−0.2 
\begin{align*}x > 10\end{align*}
x>10 
\begin{align*}y \le 7\end{align*}
y≤7 
\begin{align*}y < \frac{1}{3}\end{align*}
y<13 
\begin{align*}x \ge 10\end{align*}
x≥−10
Texas Instruments Resources
In the CK12 Texas Instruments Algebra I FlexBook, 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.
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Image Attributions
Here you'll learn how to graph linear inequalities in one variable on the coordinate plane. You'll also learn how to find their solution plane.