What if you were given a linear inequality like \begin{align*}|y| \ge -5\end{align*}? How could you graph that inequality in the coordinate plane? After completing this Concept, you'll be able to graph linear inequalities in one variable like this one.
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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*} or \begin{align*}y < mx + b\end{align*}. Linear inequalities are closely related to graphs of straight lines; recall that a straight line has the equation \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 above the line and also all the points on the line.
< The solution set is the half plane below the line.
\begin{align*}\le\end{align*} The solution set is the half plane below the line and also all the points on the line.
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*}; the solution set is the line and the half plane above the line.
This is a graph of \begin{align*}y < mx + b\end{align*}; the solution set is the half plane above the line, not including the line itself.
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*} we get a vertical line, and when we graph an equation of the type \begin{align*}y = b\end{align*} we get a horizontal line.
Example B
Graph the inequality \begin{align*}x > 4\end{align*} on the coordinate plane.
Solution
First let’s remember what the solution to \begin{align*}x > 4\end{align*} looks like on the number line.
The solution to this inequality is the set of all real numbers \begin{align*}x\end{align*} that are bigger than 4, not including 4. The solution is represented by a line.
In two dimensions, the solution still consists of all the points to the right of \begin{align*}x = 4\end{align*}, but for all possible \begin{align*}y-\end{align*}values as well. This solution is represented by the half plane to the right of \begin{align*}x = 4\end{align*}. (You can think of it as being like the solution graphed on the number line, only stretched out vertically.)
The line \begin{align*}x = 4\end{align*} is dashed because the equals sign is not included in the inequality, meaning that points on the line are not included in the solution.
Example C
Graph the inequality \begin{align*}|y| < 5\end{align*}
Solution
The absolute value inequality \begin{align*}|y| < 5\end{align*} can be re-written as \begin{align*}-5 < y < 5\end{align*}. This is a compound inequality which can be expressed as
\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*} is larger than -5 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
Vocabulary
- Inequalities of the type \begin{align*}|x|<a\end{align*} can be rewritten as “\begin{align*}-a < x < a\end{align*}.”
- Inequalities of the type \begin{align*}|x|>b\end{align*} can be rewritten as “\begin{align*}x < -b\end{align*} or \begin{align*}x > b\end{align*}.”
- Horizontal lines are defined by the equation \begin{align*}y=\end{align*} constant and vertical lines are defined by the equation \begin{align*}x= \end{align*} 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 above the line and also all the points on the line.
< The solution set is the half plane below the line.
\begin{align*}\le\end{align*} The solution set is the half plane below the line and also all the points on the line.
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.
Practice
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*}
Texas Instruments Resources
In the CK-12 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.