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# Linear Inequalities in Two Variables

## Graph inequalities like 2x + 3 < y on the coordinate plane

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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?

### Graphing Inequalities

Graphing linear inequalities is very similar to graphing lines. First, you need to change the inequality into slope-intercept form. At this point, we will have a couple of differences. If the inequality is in the form y<mx+b\begin{align*}y < mx + b\end{align*} or y>mx+b\begin{align*}y > mx + b\end{align*}, the line will be dotted or dashed because it is not a part of the solution. If the line is in the form ymx+b\begin{align*}y \le mx + b\end{align*} or ymx+b\begin{align*}y \ge mx + b\end{align*}, the line will be solid to indicate that it is included in the solution.

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 y<mx+b\begin{align*}y < mx + b\end{align*} or ymx+b\begin{align*}y \le mx + b\end{align*}, the shading will be below the line, in reference to the y\begin{align*}y-\end{align*}axis.

If the inequality is in the form y>mx+b\begin{align*}y > mx + b\end{align*} or ymx+b\begin{align*}y \ge mx + b\end{align*}, the shading will be above the line.

Let's graph the following inequalities.

1. 4x2y<10\begin{align*}4x -2y < 10\end{align*}

First, change the inequality into slope-intercept 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 y=2x5\begin{align*}y = 2x - 5\end{align*}. However, the line will be dashed because of the “greater than” sign.

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 slope-intercept 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 20<10\begin{align*}-20 < 10\end{align*}, which tells us that our shading is correct.

1. y23x+6\begin{align*}y \le - \frac{2}{3}x + 6\end{align*}

This inequality is already in slope-intercept 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 06\begin{align*}0 \le 6\end{align*}, which is true.

Let's determine the linear inequality that is graphed below.

Find the equation of the line portion just like you've done before. The given points on the line are (0, 8) and (6, 2) (from the points drawn on the graph). This means that the y\begin{align*}y-\end{align*}intercept is (0, 8). Then, using slope triangles we fall 6 and run 6 to get to (6, 2). This means the slope is 66\begin{align*}\frac{-6}{6}\end{align*} or -1. Because we have a dotted line and the shading is above, our sign will be the > sign. Putting it all together, the equation of our linear inequality is y>x+8\begin{align*}y > -x + 8\end{align*}.

\begin{align*}^*\end{align*}When finding the equation of an inequality, like above, it is easiest to find the equation in slope-intercept form. To determine which inequality sign to use, look at the shading along the y\begin{align*}y-\end{align*}axis. If the shaded y\begin{align*}y-\end{align*}values get larger, the line will be in the form or y>mx+b\begin{align*}y > mx + b\end{align*} or ymx+b\begin{align*}y \ge mx + b\end{align*}. If they get smaller, the line will be in the form y<mx+b\begin{align*}y < mx + b\end{align*} or ymx+b\begin{align*}y \le mx + b\end{align*}.

### Examples

#### Example 1

Earlier, you were asked to find which quadrant(s) into which the graph represented by the situation would fall.

To solve this taxi cab problem, we must first set up an inequality to represent the situation.

1.5x0.25y10\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.

#### Example 2

Graph 3x4y>20\begin{align*}3x - 4y > 20\end{align*}.

First, change the inequality into slope-intercept form.

3x4y4yy>20>3x+20<34x5\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 42>20\begin{align*}42 > 20\end{align*}, which is true.

#### Example 3

Graph x<1\begin{align*}x < -1\end{align*}.

To graph this line on the xy\begin{align*}x-y\end{align*} plane, recall that all vertical lines have the form x=a\begin{align*}x = a\end{align*}. Therefore, we will have a vertical dashed line at -1. Then, the shading will be to the left of the dashed line because that is where x\begin{align*}x\end{align*} will be less than the value of the line.

#### Example 4

What is the equation of the linear inequality?

Looking at the line, the y\begin{align*}y-\end{align*}intercept is (0, 1). Using a slope triangle, to count down to the next point, we would fall 4, and run backward 1. This means that the slope is 41=4\begin{align*}\frac{-4}{-1} = 4\end{align*}. The line is solid and the shading is above, so we will use the \begin{align*}\ge\end{align*} sign. Our inequality is y4x+1\begin{align*}y \ge 4x + 1\end{align*}.

### Review

Graph the following inequalities.

1. y>x5\begin{align*}y > x -5\end{align*}
2. 3x2y4\begin{align*}3x -2y \ge 4\end{align*}
3. y<3x+8\begin{align*}y < -3x + 8\end{align*}
4. x+4y16\begin{align*}x +4y \le 16\end{align*}
5. y<2\begin{align*}y < -2\end{align*}
6. y<12x3\begin{align*}y < - \frac{1}{2}x - 3\end{align*}
7. x6\begin{align*}x \ge 6\end{align*}
8. 8x+4y20\begin{align*}8x +4y \ge -20\end{align*}
9. 4x+y7\begin{align*}-4x + y \le 7\end{align*}
10. 5x3y24\begin{align*}5x -3y \ge -24\end{align*}
11. y>5x\begin{align*}y > 5x\end{align*}
12. y0\begin{align*}y \le 0\end{align*}

Determine the equation of each linear inequality below.

To see the Review answers, open this PDF file and look for section 2.11.

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### Vocabulary Language: English

TermDefinition
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: $ax + b > c, ax + b < c, ax + b \ge c$, or $ax + b \le c$.
Slope-Intercept Form The slope-intercept form of a line is $y = mx + b,$ where $m$ is the slope and $b$ is the $y-$intercept.