Graphing Solutions of Systems of Inequalities

2.14.4 Graphing Solutions of Systems of Inequalities

2.14.4 • Graphing Solutions of Systems of Inequalities

Activity

Members of a high school math club have a scavenger hunt. Three items are hidden in the park, which is a rectangle that measures 50 meters by 20 meters.

The clues are written as systems of inequalities. One system has no solutions.
The locations of the items can be narrowed down by solving the systems. A coordinate plane can be used to describe the solutions.

Can you find the hidden items? Sketch a graph to show where each item could be hidden.

You will be provided with a sheet of graph paper to help solve this scavenger hunt without the use of technology. For each clue, graph each inequality separately on the same coordinate grid.

Clue 1:

$y > 14$

$x < 10$

A blank grid of the first quadrant with an x-axis extending from 0 to 50 with a scale by 10’s. The y-axis also has a scale of 10 and extends from 0 to 30.

Compare your answer:

Graph of a system of two linear inequalities on a coordinate plane.

Clue 2:

$x + y < 20$

$x > 6$

Compare your answer:

Clue 3:

$y < - 2 x + 20$

$y < - 2 x + 10$

Compare your answer:

Clue 4:

$y \geq x + 10$

$x > y$

Compare your answer:

Reflect: When you graphed each pair of inequalities, you created a system of inequalities. Where did the possible solutions lie on the graph?

Compare your answer:

Solution Compare your answer: Your answer may vary, but here is a sample. Where the shaded regions overlap.

Are you ready for more?

Extending Your Thinking

Two non-negative numbers $x$ and $y$ satisfy $x + y \leq 1$ . Find a second inequality, also using $x$ and $y$ values greater than or equal to zero, to make a system of inequalities with exactly one solution.

Compare your answer: Your answer may vary, but here is a sample. $y \geq 1$ .

Two non-negative numbers $x$ and $y$ satisfy $x + y \leq 1$ . Find as many ways to identify the answers to this question as you can.

Compare your answer: Your answer may vary, but here is a sample. $x \geq 1$ and any inequality of the form $y \geq a x$ , where $a$ is negative.

Self Check

Which two inequalities are graphed in the system of inequalities below?

GRAPH OF A SYSTEM OF INEQUALITIES WITH SOLID VERTICAL LINE THROUGH X EQUALS 2 AND SHADING TO THE RIGHT AND DOTTED LINE WITH Y

$x>2\\y\geq3x-1\\$
$x\geq2\\y<3x-1\\$
$x\geq2\\y>3x-1\\$
$x\leq2\\y>3x-1\\$

Additional Resources

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

Solve the system by graphing:

${\begin{matrix} y \geq 2 x - 1 \\ y < x + 1 \end{matrix}$

Answer:

Step 1 - Graph the first inequality.

Graph the boundary line.
Shade in the side of the boundary line where the inequality is true.

We will graph $y \geq 2 x - 1$

We graph the line $y = 2 x - 1 .$ It is a solid line because the inequality sign is $\geq$ .

We choose $(0, 0)$ as a test point. It is a solution to $y \geq 2 x - 1$ , so we shade in above the boundary line.

Graph of an inequality where the shaded region is above the solid boundary line.

Step 2 - On the same grid, graph the second inequality.

Graph the boundary line.
Shade the side of that boundary line where the inequality is true.

We will graph $y < x + 1$ on the same grid.

We graph the line $y = x + 1$ . It is a dotted line because the inequality sign is $<$ .

Again, we use $(0, 0)$ as a test point. It is a solution so we shade in that side of the line $y = x + 1$ .

Graph of a system of two inequalities on a coordinate plane. The graph of the red line has a solution region that lies above the solid boundary line. The graph of the blue line has a solution region that lies below the dashed boundary line. The solution region for the system lies in the overlapping dark region.

Step 3 - The solution is the region where the shading overlaps.

The point where the boundary lines intersect is not a solution because it is not a solution to $y < x + 1$ .

The solution is all points in the area shaded twice, which appears as the darkest shaded region.

Step 4 - Check by choosing a test point.

We’ll use $(- 1, - 1)$ as a test point.

Is $(- 1, 1)$ a solution to $y \geq 2 x - 1$ ?

$- 1 \overset{?}{\geq} 2 (- 1) - 1$

$- 1 \geq - 3$

$✓$

Is $(- 1, - 1)$ a solution to $y < x + 1$ ?

$- 1 \overset{?}{\geq} - 1 + 1$

$- 1 < 0$

$✓$

The region containing $(- 1, - 1)$ is the solution to this system.

Try it

Solve a System of Linear Inequalities by Graphing

Solve the system by graphing

${\begin{matrix} y < 3 x + 2 \\ y > - x - 1 \end{matrix}$

Compare your answer: Here is how to solve the system by graphing.

Where the shaded regions overlap.

Graph of a system of two inequalities on a coordinate plane. The graph of the red line has a solution region that lies below the dashed boundary line. The graph of the blue line has a solution region that lies above the dashed boundary line. The solution region for the system lies in the overlapping dark region to the right.