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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 (3.3)

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?

### Guidance

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

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

#### Example A

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

Solution: 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 \begin{align*}y = 2x - 5\end{align*} like in the Graph a Line in Slope-Intercept Form concept. 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 \begin{align*}-20 < 10\end{align*}, which tells us that our shading is correct.

#### Example B

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

Solution: 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 \begin{align*}0 \le 6\end{align*}, which is true.

#### Example C

Determine the linear inequality that is graphed below.

Solution: Find the equation of the line portion just like you did in the Finding the Equation of a Line in Slope-Intercept 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*}intercept is (0, 8). Then, using slope triangles we fall 6 and run 6 to get to (6, 2). This means the slope is \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 \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 \begin{align*}y-\end{align*}axis. If the shaded \begin{align*}y-\end{align*}values get larger, the line will be in the form or \begin{align*}y > mx + b\end{align*} or \begin{align*}y \ge mx + b\end{align*}. If they get smaller, the line will be in the form \begin{align*}y < mx + b\end{align*} or \begin{align*}y \le mx + b\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?

1. First, change the inequality into slope-intercept 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*}, which is true.

2. To graph this line on the \begin{align*}x-y\end{align*} plane, recall that all vertical lines have the form \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 \begin{align*}x\end{align*} will be less than the value of the line.

3. Looking at the line, the \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 \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 \begin{align*}y \ge 4x + 1\end{align*}.

### Practice

Graph the following inequalities.

1. \begin{align*}y > x -5\end{align*}
2. \begin{align*}3x -2y \ge 4\end{align*}
3. \begin{align*}y < -3x + 8\end{align*}
4. \begin{align*}x +4y \le 16\end{align*}
5. \begin{align*}y < -2\end{align*}
6. \begin{align*}y < - \frac{1}{2}x - 3\end{align*}
7. \begin{align*}x \ge 6\end{align*}
8. \begin{align*}8x +4y \ge -20\end{align*}
9. \begin{align*}-4x + y \le 7\end{align*}
10. \begin{align*}5x -3y \ge -24\end{align*}
11. \begin{align*}y > 5x\end{align*}
12. \begin{align*}y \le 0\end{align*}

Determine the equation of each linear inequality below.

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