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# Linear Inequalities in Two Variables

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

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Practice Linear Inequalities in Two Variables
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Graphing Linear Inequalities

Did you know that you can use an inequality to describe a real-world situation? Take a look at this dilemma.

You’re going to a party! You’re supposed to bring sodas and chips but you only have $20 to spend. Sodas cost$1.50 per bottle and chips cost 2.50 per bag. How many of each can you buy? This situation can be modeled with a linear inequality. Here, you will learn how to solve this inequality by graphing. ### Guidance Listing solutions to single variable inequalities is useful but, because there are infinitely many solutions, it is impossible to show the entire solution set with a list. For that reason, we use number lines. So when \begin{align*}x<2\end{align*}, we show it like this. With the less than \begin{align*}(<)\end{align*}, greater than \begin{align*}(>)\end{align*}, and not equal to \begin{align*}(\ne)\end{align*} symbols, we use an open circle because, although the solution set is infinitely close to the endpoint, the endpoint itself is not actually a solution. With the less than or equal to \begin{align*}(\le)\end{align*} and greater than or equal to \begin{align*}(\ge)\end{align*} symbols, the endpoint is a solution so we use a closed circle. Just as we graphed linear equations, we can also graph linear inequalities. We will graph the linear inequalities using slope-intercept form. As the circle on a number line marks the end of the solution set of a single variable inequality, so the line on the coordinate plane will mark the boundary of the solution set of a linear inequality. The solution set will be on one side of the line or the other. We will take a test point to figure out which side makes the inequality true and then shade that half of the coordinate plane to indicate the solution set. With the less than \begin{align*}(<)\end{align*}, greater than \begin{align*}(>)\end{align*}, and not equal to \begin{align*}(\ne)\end{align*} symbols, we will use a dotted line instead of an open circle because, although the solution set is infinitely close to the line, the points on the line itself are not actually solutions. With the less than or equal to \begin{align*}(\le)\end{align*} and greater than or equal to \begin{align*}(\ge)\end{align*} symbols, the points on the line are solutions so we use a solid line. Take a look at this situation. Graph the solution set of the inequality \begin{align*}y > x+3\end{align*}. Graph using \begin{align*}m=1, b=3\end{align*}. Use a dotted line because the symbol is \begin{align*} > \end{align*}. Now, the solution set is on one side of the line or the other. In order to determine which side, we will just try a point that is not on the line itself. Try, for example, (1, 1). Does the point make the inequality \begin{align*}y>x+3\end{align*} true? \begin{align*}1>1+3?\end{align*} No! The solution set must be on the other side of the line. As you can see in the graph, we shaded the opposite side. We can graph any inequality in this way. First, graph the equation of the line. Then check if it is a solid or dashed line. Then shade above or below the line based on the inequality symbol. Write these steps down in your notebook. Answer each question about graphing inequalities. #### Example A True or false. If the inequality is less than, then a graph will show that the area below a dotted line is shaded. Solution: True #### Example B True or false. If the inequality is greater than or equal to, then a graph will show that the area above a dotted line is shaded. Solution: False. The line will be a solid line. #### Example C True or false. With a system of linear inequalities, the area that is shaded must be shared by both inequalities. Solution: True Now let's go back to the dilemma from the beginning of the Concept. This situation can be modeled with a linear inequality. Let \begin{align*}s\end{align*} equal the number of sodas you buy and \begin{align*}c\end{align*} the number of chips that you buy. The inequality is \begin{align*} 1.5s + 2.5c \le 20 \end{align*} because the cost of the sodas plus the cost of the chips must not be more than20.

Graph the inequality and shade the correct region. Find 5 combinations of sodas and chips that you could buy by looking at the ordered pairs within the solution on the graph.

You can buy any combination that is one of the ordered pairs in the shaded region. Possible answers are (11, 1) (9, 2) (7, 3) (6, 4) etc.

### Guided Practice

Here is one for you to try on your own.

Write the inequality represented by this graph.

Solution

First, notice that the area shaded is greater than the line. Also, it is a solid line, so we know that the solution to the inequality is above the line and includes the values on the line.

The slope of the line is \begin{align*}2\end{align*}. The y-intercept is \begin{align*}-3\end{align*}.

\begin{align*}y \ge 2x-3\end{align*}

This is our answer.

### Explore More

Directions: Write an inequality for each graph.

1. .
1. .
1. .
1. .

Directions: Graph the following inequalities on the coordinate plane.

1. \begin{align*}y<2x+1\end{align*}
2. \begin{align*}y \ge 3x-2\end{align*}
3. \begin{align*}y \ge -1/2x\end{align*}
4. \begin{align*}y \le 1/4x + 2\end{align*}
5. \begin{align*}y<-2x\end{align*}
6. \begin{align*}y \le 4\end{align*}

Directions: Answer each question true or false.

1. You can't shade less than a vertical line.
2. A dotted line can only be used in an inequality with greater than.
3. A dotted line is used when the inequality sign does not include an equals.
4. You can shade less than or greater than a horizontal line.
5. You can graph a linear inequality in two variables.

### Vocabulary Language: English

Cartesian Plane

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.
inequality

inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.
Linear Inequality

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

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.