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# 6.12: Linear Inequalities in Two Variables

Difficulty Level: At Grade Created by: CK-12
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Practice Linear Inequalities in Two Variables

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What if you were given a linear inequality like 2x3y5\begin{align*}2x - 3y \le 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 two variables like this one.

### Guidance

The general procedure for graphing inequalities in two variables is as follows:

1. Re-write the inequality in slope-intercept form: y=mx+b\begin{align*}y=mx+b\end{align*}. Writing the inequality in this form lets you know the direction of the inequality.
2. Graph the line of the equation y=mx+b\begin{align*}y=mx+b\end{align*} using your favorite method (plotting two points, using slope and y\begin{align*}y-\end{align*}intercept, using y\begin{align*}y-\end{align*}intercept and another point, or whatever is easiest). Draw the line as a dashed line if the equals sign is not included and a solid line if the equals sign is included.
3. Shade the half plane above the line if the inequality is “greater than.” Shade the half plane under the line if the inequality is “less than.”

#### Example A

Graph the inequality y2x3\begin{align*}y \ge 2x-3\end{align*}.

Solution

The inequality is already written in slope-intercept form, so it’s easy to graph. First we graph the line y=2x3\begin{align*}y=2x-3\end{align*}; then we shade the half-plane above the line. The line is solid because the inequality includes the equals sign.

#### Example B

Graph the inequality 5x2y>4\begin{align*}5x-2y>4\end{align*}.

Solution

First we need to rewrite the inequality in slope-intercept form:

2yy>5x+4<52x2\begin{align*} -2y & > -5x +4\\ y & < \frac{5}{2}x - 2\end{align*}

Notice that the inequality sign changed direction because we divided by a negative number.

To graph the equation, we can make a table of values:

x\begin{align*}x\end{align*} y\begin{align*}y\end{align*}
-2 52(2)2=7\begin{align*}\frac{5}{2}(-2)-2=-7\end{align*}
0 52(0)2=2\begin{align*}\frac{5}{2}(0)-2=-2\end{align*}
2 52(2)2=3\begin{align*}\frac{5}{2}(2)-2=3\end{align*}

After graphing the line, we shade the plane below the line because the inequality in slope-intercept form is less than. The line is dashed because the inequality does not include an equals sign.

Solve Real-World Problems Using Linear Inequalities

In this section, we see how linear inequalities can be used to solve real-world applications.

#### Example C

A retailer sells two types of coffee beans. One type costs $9 per pound and the other type costs$7 per pound. Find all the possible amounts of the two different coffee beans that can be mixed together to get a quantity of coffee beans costing 8.50 or less. Solution Let x=\begin{align*}x =\end{align*} weight of9 per pound coffee beans in pounds.

Let y=\begin{align*}y =\end{align*} weight of 7 per pound coffee beans in pounds. The cost of a pound of coffee blend is given by 9x+7y\begin{align*}9x + 7y\end{align*}. We are looking for the mixtures that cost8.50 or less. We write the inequality 9x+7y8.50\begin{align*}9x+7y \le 8.50\end{align*}.

Since this inequality is in standard form, it’s easiest to graph it by finding the x\begin{align*}x-\end{align*} and y\begin{align*}y-\end{align*}intercepts. When x=0\begin{align*}x=0\end{align*}, we have 7y=8.50\begin{align*}7y=8.50\end{align*} or y=8.5071.21\begin{align*}y=\frac{8.50}{7} \approx 1.21\end{align*}. When y=0\begin{align*}y=0\end{align*}, we have 9x=8.50\begin{align*}9x=8.50\end{align*} or x=8.5090.94\begin{align*}x=\frac{8.50}{9} \approx 0.94\end{align*}. We can then graph the line that includes those two points.

Now we have to figure out which side of the line to shade. In y\begin{align*}y-\end{align*}intercept form, we shade the area below the line when the inequality is “less than.” But in standard form that’s not always true. We could convert the inequality to y\begin{align*}y-\end{align*}intercept form to find out which side to shade, but there is another way that can be easier.

The other method, which works for any linear inequality in any form, is to plug a random point into the inequality and see if it makes the inequality true. Any point that’s not on the line will do; the point (0, 0) is usually the most convenient.

In this case, plugging in 0 for x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} would give us 9(0)+7(0)8.50\begin{align*}9(0)+7(0) \le 8.50\end{align*}, which is true. That means we should shade the half of the plane that includes (0, 0). If plugging in (0, 0) gave us a false inequality, that would mean that the solution set is the part of the plane that does not contain (0, 0).

Notice also that in this graph we show only the first quadrant of the coordinate plane. That’s because weight values in the real world are always nonnegative, so points outside the first quadrant don’t represent real-world solutions to this problem.

Watch this video for help with the Examples above.

### Vocabulary

• 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

Julius has a job as an appliance salesman. He earns a commission of $60 for each washing machine he sells and$130 for each refrigerator he sells. How many washing machines and refrigerators must Julius sell in order to make 1000 or more in commissions? Solution Let x=\begin{align*}x =\end{align*} number of washing machines Julius sells. Let y=\begin{align*}y =\end{align*} number of refrigerators Julius sells. The total commission is 60x+130y\begin{align*}60x + 130y\end{align*}. We’re looking for a total commission of1000 or more, so we write the inequality 60x+130y1000\begin{align*}60x+130y \ge 1000\end{align*}.

Once again, we can do this most easily by finding the x\begin{align*}x-\end{align*} and y\begin{align*}y-\end{align*}intercepts. When x=0\begin{align*}x=0\end{align*}, we have 130y=1000\begin{align*}130y=1000\end{align*}, or y=1000307.69\begin{align*}y=\frac{1000}{30} \approx 7.69\end{align*}. When y=0\begin{align*}y=0\end{align*}, we have 60x=1000\begin{align*}60x=1000\end{align*}, or x=10006016.67\begin{align*}x=\frac{1000}{60} \approx 16.67\end{align*}.

We draw a solid line connecting those points, and shade above the line because the inequality is “greater than.” We can check this by plugging in the point (0, 0): selling 0 washing machines and 0 refrigerators would give Julius a commission of $0, which is not greater than or equal to$1000, so the point (0, 0) is not part of the solution; instead, we want to shade the side of the line that does not include it.

Notice also that we show only the first quadrant of the coordinate plane, because Julius’s commission should be non-negative.

### Practice

Graph the following inequalities on the coordinate plane.

1. y4x+3\begin{align*}y \le 4x+3\end{align*}
2. y>x26\begin{align*}y > -\frac{x}{2}-6\end{align*}
3. 3x4y12\begin{align*}3x-4y \ge 12\end{align*}
4. x+7y<5\begin{align*}x+7y < 5\end{align*}
5. 6x+5y>1\begin{align*}6x+5y>1\end{align*}
6. y+54x+10\begin{align*}y+5 \le -4x+10\end{align*}
7. x12y5\begin{align*}x-\frac{1}{2}y \ge 5\end{align*}
8. 6x+y<20\begin{align*}6x+y < 20\end{align*}
9. 30x+5y<100\begin{align*}30x+5y < 100\end{align*}
10. Remember what you learned in the last chapter about families of lines.
1. What do the graphs of y>x+2\begin{align*}y > x+2\end{align*} and y<x+5\begin{align*}y < x+5\end{align*} have in common?
2. What do you think the graph of x+2<y<x+5\begin{align*}x+2 < y < x+5\end{align*} would look like?
11. How would the answer to problem 6 change if you subtracted 2 from the right-hand side of the inequality?
12. How would the answer to problem 7 change if you added 12 to the right-hand side?
13. How would the answer to problem 8 change if you flipped the inequality sign?
14. A phone company charges 50 cents per minute during the daytime and 10 cents per minute at night. How many daytime minutes and nighttime minutes could you use in one week if you wanted to pay less than 20? 15. Suppose you are graphing the inequality y>5x\begin{align*}y > 5x\end{align*}. 1. Why can’t you plug in the point (0, 0) to tell you which side of the line to shade? 2. What happens if you do plug it in? 3. Try plugging in the point (0, 1) instead. Now which side of the line should you shade? 16. A theater wants to take in at least2000 for a certain matinee. Children’s tickets cost $5 each and adult tickets cost$10 each.
1. If x\begin{align*}x\end{align*} represents the number of adult tickets sold and y\begin{align*}y\end{align*} represents the number of children’s tickets, write an inequality describing the number of tickets that will allow the theater to meet their minimum take.
2. If 100 children’s tickets and 100 adult tickets have already been sold, what inequality describes how many more tickets of both types the theater needs to sell?
3. If the theater has only 300 seats (so only 100 are still available), what inequality describes the maximum number of additional tickets of both types the theater can sell?

### 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.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

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.

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