4.7 Graphing Systems of Linear Inequalities

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4.7.1 Determine whether an ordered pair is a solution of a system of linear inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.

SYSTEM OF LINEAR INEQUALITIES

Two or more linear inequalities grouped together form a system of linear inequalities.

A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.

\Bigg\{ \begin{align*} &x+4y≥10 \\ &3x-2y<12 \end{align*}

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs $(x, y)$ that make both inequalities true.

SOLUTIONS OF A SYSTEM OF LINEAR INEQUALITIES

Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.

The solution of a system of linear inequalities is shown as a shaded region in the $x$, $y$ coordinate system that includes all the points whose ordered pairs make the inequalities true.

To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

Example 1

Determine whether the ordered pair is a solution to the system \Bigg\{ \begin{align*} &x+4y≥10 \\ &3x-2y<12 \end{align*}

• $(-2, 4)$
• $(3, 1)$
Solution

Part 1– Is the ordered pair $(-2, 4)$ a solution?

We substitute $\textcolor{red}{x=-2}$ and $\textcolor{blue}{y=4}$ into both inequalities.

 \begin{align*} \textcolor{red}{x}+4\textcolor{blue}{y}&≥10 \\ \textcolor{red}{-2}+4(\textcolor{blue}{4})&≥10 \\ 14&≥10 \text{ true} \end{align*} \begin{align*} 3\textcolor{red}{x}-2\textcolor{blue}{y}&<12 \\ 3(\textcolor{red}{-2})-2(\textcolor{blue}{4})&<12 \\ -14&<12 \text{ true} \end{align*}

The ordered pair $(-2, 4)$ made both inequalities true. Therefore $(-2, 4)$ is a solution to this system.

Part 2– Is the ordered pair $(3, 1)$ a solution?

We substitute $\textcolor{red}{x=3}$ and $\textcolor{blue}{y=1}$ into both inequalities.

 \begin{align*} \textcolor{red}{x}+4\textcolor{blue}{y}&≥10 \\ \textcolor{red}{3}+4(\textcolor{blue}{1})&≥10 \\ 7&≥10 \text{ false} \end{align*} \begin{align*} 3\textcolor{red}{x}-2\textcolor{blue}{y}&<12 \\ 3(\textcolor{red}{3})-2(\textcolor{blue}{1})&<12 \\ 7&<12 \text{ true} \end{align*}

The ordered pair $(3, 1)$ made one inequality true, but the other one false. Therefore $(3, 1)$ is not a solution to this system.

4.7.2 Solve a System of Linear Inequalities by Graphing

The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.

Example 2

Solve the system by graphing: \Bigg\{ \begin{align*} y&≥2x-1 \\ y&<x+1 \end{align*}

Solution

HOW TO: Solve a system of linear inequalities by graphing.

1. Graph the first inequality.
• Graph the boundary line.
• Shade in the side of the boundary line where the inequality is true.
2. On the same grid, graph the second inequality.
• Graph the boundary line.
• Shade in the side of that boundary line where the inequality is true.
3. The solution is the region where the shading overlaps.
4. Check by choosing a test point.

Example 3

Solve the system by graphing: \Bigg\{ \begin{align*} &x-y>3 \\ &y<-\frac{1}{5}x+4 \end{align*}

Solution

The point of intersection of the two lines is not included as both boundary lines were dashed. The solution is the area shaded twice—which appears as the darkest shaded region.

Example 4

Solve the system by graphing: \Bigg\{ \begin{align*} &x-2y<5 \\ &y>-4 \end{align*}

Solution

The point $(0, 0)$ is in the solution and we have already found it to be a solution of each inequality. The point of intersection of the two lines is not included as both boundary lines were dashed.

The solution is the area shaded twice—which appears as the darkest shaded region.

Systems of linear inequalities where the boundary lines are parallel might have no solution. We’ll see this in the next example.

Example 5

Solve the system by graphing: \Bigg\{ \begin{align*} &4x+3y≥12 \\ &y<-\frac{4}{3}x+1 \end{align*}

Solution

There is no point in both shaded regions, so the system has no solution.

Some systems of linear inequalities where the boundary lines are parallel will have a solution. We’ll see this in the next example.

Example 6

Solve the system by graphing: \Bigg\{ \begin{align*} &y>\frac{1}{2}x-4 \\ &x-2y<-4 \end{align*}

Solution

No point on the boundary lines is included in the solution as both lines are dashed.

The solution is the region that is shaded twice which is also the solution to $x-2y<-4$.

4.7.3 Solve Applications of Systems of Inequalities

The first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so we add inequalities to the system as additional requirements.

Christy sells her photographs at a booth at a street fair. At the start of the day, she wants to have at least $25$ photos to display at her booth. Each small photo she displays costs her $\$4$and each large photo costs her$\$10$. She doesn’t want to spend more than $\$200$on photos to display. • Write a system of inequalities to model this situation. • Graph the system. • Could she display 10 small and 20 large photos? • Could she display 20 large and 10 small photos? Solution Part 1 Part 2 Since$x≥0$and$y≥0$(both are greater than or equal to) all solutions will be in the first quadrant. As a result, our graph shows only quadrant one. The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly-shaded section are included in the solution as are the points on the$x$-axis from$(25, 0)$to$(55, 0)$. Part 3 To determine if$10$small and$20$large photos would work, we look at the graph to see if the point$(10, 20)$is in the solution region. We could also test the point to see if it is a solution of both equations.It is not, so Christy would not display$10$small and$20$large photos. Part 4 To determine if$20$small and$10$large photos would work, we look at the graph to see if the point$(20, 10)$is in the solution region. We could also test the point to see if it is a solution of both equations. It is, so Christy could choose to display$20$small and$10$large photos. Notice that we could also test the possible solutions by substituting the values into each inequality. When we use variables other than$x$and$y$to define an unknown quantity, we must change the names of the axes of the graph as well. Example 8 Omar needs to eat at least$800$calories before going to his team practice. All he wants is hamburgers and cookies, and he doesn’t want to spend more than$\$5$. At the hamburger restaurant near his college, each hamburger has $240$ calories and costs $\$1.40$. Each cookie has$160$calories and costs$\$0.50$.

• Write a system of inequalities to model this situation.
• Graph the system.
• Could he eat $3$ hamburgers and $1$ cookie?
• Could he eat $2$ hamburgers and $4$ cookies?
Solution

Part 1

Part 2

Since $h>=0$ and $c>=0$ (both are greater than or equal to) all solutions will be in the first quadrant. As a result, our graph shows only quadrant one.

The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly shaded section are included in the solution as are the points on the $x$-axis from $(5, 0)$ to $(10, 0)$.

Part 3

To determine if $3$ hamburgers and $1$ cookie would meet Omar’s criteria, we see if the point $(3, 1)$ is in the solution region. It is, so Omar might choose to eat $3$ hamburgers and $1$ cookie.

Part 4

To determine if $2$ hamburgers and $4$ cookies would meet Omar’s criteria, we see if the point $(2, 4)$ is in the solution region. It is, Omar might choose to eat $2$ hamburgers and $4$ cookies.

We could also test the possible solutions by substituting the values into each inequality.