2.11: Graphing Inequalties in Two Variables
Xpress Taxi Service charges $1.50 per minute traveled minus $0.25 per minute spent in stopped traffic. You only have $10 in your wallet, so that is the maximum amount you can spend on your ride. In which quadrant(s) would the graph represented by this situation fall?
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Khan Academy: Graphing linear inequalities in two variables 2
Guidance
Graphing linear inequalities is very similar to graphing lines. First, you need to change the inequality into slopeintercept form. At this point, we will have a couple of differences. If the inequality is in the form \begin{align*}y < mx + b\end{align*}
The second difference is the shading. Because these are inequalities, not just the line is the solution. Depending on the sign, there will be shading above or below the line. If the inequality is in the form \begin{align*}y < mx + b\end{align*}
If the inequality is in the form \begin{align*}y > mx + b\end{align*}
Example A
Graph \begin{align*}4x 2y < 10\end{align*}
Solution: First, change the inequality into slopeintercept form. Remember, that if you have to divide or multiply by a negative number, you must flip the inequality sign.
Now, graph the inequality as if it was a line. Plot \begin{align*}y = 2x  5\end{align*}
Now, we need to determine the shading. You can use one of two methods to do this. The first way is to use the graphs and forms from above. The equation, in slopeintercept form, matches up with the purple dashed line and shading. Therefore, we should shade above the dashed blue line.
The alternate method would be to test a couple of points to see if they work. If a point is true, then the shading is over that side of the line. If we pick (5, 0), the inequality yields \begin{align*}20 < 10\end{align*}
Example B
Graph \begin{align*}y \le  \frac{2}{3}x + 6\end{align*}
Solution: This inequality is already in slopeintercept form. So, graph the line, which will be solid, and then determine the shading. Looking at the example graphs above, this inequality should look like the red inequality, so shade below the line.
Test a point to make sure our shading is correct. An easy point in the shaded region is (0, 0). Plugging this into the inequality, we get \begin{align*}0 \le 6\end{align*}
Example C
Determine the linear inequality that is graphed below.
Solution: Find the equation of the line portion just like you did in the Find the Equation of a Line in SlopeIntercept Form concept. The given points on the line are (0, 8) and (6, 2) (from the points drawn on the graph). This means that the \begin{align*}y\end{align*}
\begin{align*}^*\end{align*}
Intro Problem Revisit To solve this taxi cab problem, we must first set up an inequality to represent the situation.
\begin{align*}1.5x  0.25y \le 10\end{align*}
You can't travel a negative number of miles or sit in traffic for a negative number of minutes. Therefore both x and y must have zero or positive values. When both x and y are positive, the graph occurs in the first quadrant only. Graph the function to check this answer.
Guided Practice
1. Graph \begin{align*}3x  4y > 20\end{align*}
2. Graph \begin{align*}x < 1\end{align*}
3. What is the equation of the linear inequality?
Answers
1. First, change the inequality into slopeintercept form.
\begin{align*}3x 4y & > 20\\
4y & > 3x + 20\\
y & < \frac{3}{4}x 5\end{align*}
Now, we need to determine the type of line and shading. Because the sign is “<,” the line will be dashed and we will shade below.
Test a point in the shaded region to make sure we are correct. If we test (6, 6) in the original inequality, we get \begin{align*}42 > 20\end{align*}
2. To graph this line on the \begin{align*}xy\end{align*}
3. Looking at the line, the \begin{align*}y\end{align*}
Practice
Graph the following inequalities.

\begin{align*}y > x 5\end{align*}
y>x−5 
\begin{align*}3x 2y \ge 4\end{align*}
3x−2y≥4 
\begin{align*}y < 3x + 8\end{align*}
y<−3x+8 
\begin{align*}x +4y \le 16\end{align*}
x+4y≤16 
\begin{align*}y < 2\end{align*}
y<−2 
\begin{align*}y <  \frac{1}{2}x  3\end{align*}
y<−12x−3 
\begin{align*}x \ge 6\end{align*}
x≥6 
\begin{align*}8x +4y \ge 20\end{align*}
8x+4y≥−20 
\begin{align*}4x + y \le 7\end{align*}
−4x+y≤7 
\begin{align*}5x 3y \ge 24\end{align*}
5x−3y≥−24 
\begin{align*}y > 5x\end{align*}
y>5x 
\begin{align*}y \le 0\end{align*}
y≤0
Determine the equation of each linear inequality below.
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Term  Definition 

Cartesian Plane  The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. 
Linear Inequality  Linear inequalities are inequalities that can be written in one of the following four forms: , or . 
SlopeIntercept Form  The slopeintercept form of a line is where is the slope and is the intercept. 
Image Attributions
Here you'll learn how to graph a linear inequality on the xy plane.