Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

Here is a graph that represents solutions to the equation $x - y = 5$ .

Using this graph as a guide, sketch a quick graph that represents the solutions to each of the inequalities in questions 1 – 4. Use the graphing tool or technology outside the course. Graph the equation that represents each scenario using the Desmos tool provided.

The graph of the line x minus y equals 5 on a coordinate grid displays the line crosses the y-axis at negative five and intersects the x-axis at 5.

1.

$x - y < 5$

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Compare your answer:

The graph of the inequality representing x minus y less than 5 on a coordinate grid displays the line as dashed and the shaded region located above the line and to the left.

2.

$x - y \leq 5$

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Compare your answer:

The graph of the inequality representing x minus y less than or equal to 5 on a coordinate grid displays the line as solid and the shaded region located above the line and to the left.

3.

$x - y > 5$

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Compare your answer:

The graph of the inequality representing x minus y greater than 5 on a coordinate grid displays the line as dashed and the shaded region located below the line and to the right.

4.

$x - y \geq 5$

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Compare your answer:

The graph of the inequality representing x minus y greater than or equal to 5 on a coordinate grid displays the line as solid and the shaded region located below the line and to the right.

For each graph in questions 5 – 8, write an inequality whose solutions are represented by the shaded part of the graph.

5.

The graph on the coordinate plane shows a region with a vertical boundary line that is dashed and shaded to the right. The vertical line intersects the x-axis at (3, 0).

Compare your answer:

$x > 3$

6.

The graph on the coordinate plane shows a region shaded below a horizontal boundary line that is solid. The horizontal line has a y-intercept at (0, negative 2).

Compare your answer:

$y \leq - 2$

7.

The graph on the coordinate plane displays an inequalilty that is shaded above a dashed boundary line that passes through the points (negative 4, negative 4), (0, 0) and (5, 5).

Compare your answer:

$y > x$

8.

The graph on the coordinate plane displays an inequalilty that is shaded below a solid boundary line that passes through the points (negative 4, negative 4), (0, 0) and (5, 5).

Compare your answer:

$y \leq x$ or $x \geq y$

Are you ready for more?

Extending Your Thinking

The graph on the coordinate plane displays an inequalilty that is shaded above a dashed boundary line.

This is a graph of the inequality $x - 2 y < 3$ . The points $(7, 3)$ and $(7, 5)$ are both in the solution region for this inequality. Use this information to answer questions 1 – 5.

1.

Compute $x - 2 y$ for both of these points.

Compare your answer:

1 and -3

2.

Which point comes closest to satisfying the equation $x - 2 y = 3$ ? That is, for which $(x, y)$ pair is $x - 2 y$ closest to 3?

Compare your answer:

$(7, 3)$

3.

The points $(3, 2)$ and $(5, 2)$ are also in the solution region. Which of these points comes closest to satisfying the equation $x - 2 y = 3$ ?

Compare your answer:

Compare your answer: $(5, 2)$ is closer because $(3, 2) = (3) - (2) (2) = - 1$ $(5, 2) = (5) - (2) (2) = 1$

1 is closer to 3 than -1, so the point that comes closest to satisfying the equation is $(5, 2)$ .

4.

Find a point in the solution region that comes even closer to satisfying the equation $x - 2 y = 3$ . What is the value of $x - 2 y$ ?

Compare your answer: Your answer may vary, but here is a sample. $6 - (2) (2) = 2$

5.

For the points $(5, 2)$ and $(7, 3)$ , $x - 2 y = 1$ . Find another point in the solution region for which $x - 2 y = 1$ .

Compare your answer: Your answer may vary, but here is a sample. $(3, 1)$ Any point on the line $y = \frac{1}{2} x - \frac{1}{2}$ will work.

Use the point $(5, 3)$ to answer questions 6 and 7.

Use the point $(5, 3)$ to answer question 6.

6.

Find the value of $x - 2 y$ .

$x = 5$

$y = 3$

$(5) - 2 (3)$

$5 - 6 = - 1$

7.

Find two other points that give the same answer.

Compare your answer: Your answer may vary, but here are some samples.

$(3, 2)$ (from previous work), $(1, 1)$
Any point on the line $y = \frac{1}{2} x + \frac{1}{2}$ will work.

Self Check

Which graph represents the inequality

y \leq 3x + 2

?

Additional Resources

Recognize the Relation Between the Solutions of an Inequality and its Graph

Now, we will look at how the solutions of an inequality relate to its graph. Let’s think about the number line shown previously again. The point $x = 3$ separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See the figure below.

A number line from negative 5 to 5 with arrows above it. On the number line, there is an open parenthesis at 3 with the numbers to the right of 3 shaded. Above the number line is a set of arrows. One arrow points left from 3 and is labeled numbers less than 3. The other arrow above the number line points right from 3 and is labeled numbers greater than 3.

The solution of $x > 3$ is the shaded part of the number line to the right of $x = 3$

Similarly, the line $y = x + 4$ separates the plane into two regions. On one side of the line are points with $y < x + 4$ . On the other side of the line are the points with $y > x + 4$ . We call the line $y = x + 4$ a boundary line.

Boundary Line

The line with equation $A x + B y = C$ is the boundary line that separates the region where $A x + B y > C$ from the region where $A x + B y < C$ .

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not $a$ is included in the solution:

Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or not it the line is included in the solution.

$A x + B y < C$	$A x + B y \leq C$
$A x + B y > C$	$A x + B y \geq C$
Boundary line is $A x + B y = C$	Boundary line is $A x + B y = C$
Boundary line is not included in solution.	Boundary line is included in solution.
Boundary line is dashed.	Boundary line is solid.

Now, let’s take a look at what we found in the example above. We’ll start by graphing the line $y = x + 4$ , and then we’ll plot the five points we tested, as shown in the graph. See the figure below.

The graph of y equals x plus four on a coordinate plane. Five additional points are plotted on the plane at (0, 0), (1, 6), (2, 6), (negative 5, negative 15), and (negative 8, 12).

In the previous example, we found that some of the points were solutions to the inequality $y > x + 4$ and some were not.

Which of the points we plotted are solutions to the inequality $y > x + 4$ ?

The points $(1, 6)$ and $(- 8, 12)$ are solutions to the inequality $y > x + 4$ . Notice that they are both on the same side of the boundary line $y > x + 4$ .

The two points $(0, 0)$ and $(- 5, - 15)$ are on the other side of the boundary line $y = x + 4$ , and they are not solutions to the inequality $y > x + 4$ . For those two points. $y < x + 4$ .

What about the point $(2, 6)$ ? The point $(2, 6)$ is a solution to the boundary line equation $y = x + 4$ , because $6 = 2 + 4$ . However, $(2, 6)$ is not a solution to the inequality, because the boundary line is not included in the solution to the inequality $y > x + 4$ .

Let's take another point above the boundary line and test whether or not it is a solution to the inequality $y > x + 4$ . The point $(0, 10)$ clearly looks to above the boundary line, doesn't it? Is it a solution to the inequality?

$y > x + 4$

$10 > 0 + 4$

$10 > 4$

So, $(0, 10)$ is a solution to $y > x + 4$ .

Any point you choose above the boundary line is a solution to the inequality $y > x + 4$ . All points above the boundary line are solutions.

Similarly, all points below the boundary line, the side with $(0, 0)$ and $(- 5, - 15)$ , are not solutions to $y > x + 4$ , as shown in the figure below.

The graph of y equals x plus four on a coordinate plane. Six additional points are plotted on the plane at (0, 10), (0, 0), (1, 6), (2, 6), (negative 5, negative 15), and (negative 8, 12). The half plane above the line and to the left is labeled as y greater than x plus 4. The region below the line and to the right is labeled as y less than x plus 4.

The graph of the inequality $y > x + 4$ is shown in below.

The line $y = x + 4$ divides the plane into two regions. The shaded side shows the solutions to the inequality $y > x + 4$ .

The points on the boundary line, those where $y = x + 4$ , are not solutions to the inequality $y > x + 4$ , so the line itself is not part of the solution. We show that by making the line dashed, not solid.