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

Activity

A close-up of a colorful trail mix containing dried apricots, raisins, white chocolate chips, walnuts, hazelnuts, dried cranberries, and other mixed nuts and dried fruits.

For questions 1 – 5, use the following scenario to graph an inequality from a table:

Andre is making trail mix with dried fruit and mixed nuts. Dried fruit costs $3 per pound and mixed nuts cost $6 per pound. He wants to spend no more than $54 on the trail mix. The inequality $y \leq - 2 x + 18$ represents the number of pounds of dried fruit, $y$ , he can buy for $x$ pounds of nuts.

1.

The related equation $y = - 2 x + 18$ represents the boundary line of the inequality. Complete the table of values to find points on the boundary line of the inequality.

$x$	$y$
0
2
4
6

Compare your answer:

$x$	$y$
0	18
2	14
4	10
6	6

2.

Use the graphing tool or technology outside the course. Graph the data from your table of values that represents this scenario using the Desmos tool below.

Access multimedia content

Compare your answer:

Scatter plot titled Trail Mix showing a negative correlation between mixed nuts and dried fruit. The weight of mixed nuts in pounds is represented on the x-axis. The weight of the dried fruit in pounds is represented on the y-axis.

3.

Is the boundary line for the inequality $y \leq - 2 x + 18$ a solid line or a dashed line? Explain how you know.

Compare your answer:

The line is solid because the inequality $\leq$ means the points on the line are included as solutions.

4.

Is the point $(0, 0)$ a solution to the inequality?

Compare your answer:

Yes, $0 \leq - 2 (0) + 18$ , $0 \leq 18$ .

5.

Use the graph from number 2. Enter the inequality $y \leq - 2 x + 18$ to represent the data. Notice the shading for the solution region of this inequality.

Compare your answer:

A linear inequality graph titled Trail Mix shows pounds of dried fruit on the y-axis and the pounds of mixed nuts on the x-axis. The boundary line slopes down from 18 pounds dried fruit and 0 mixed nuts, to 0 pounds dried fruit and 9 pounds mixed nuts.

In problems 6–10, use the data from the table to graph the inequality and write an inequality with two variables.

$x$	$y$
-2	-5
0	-6
2	1
3	5

6.

Use the graphing tool or technology outside the course. Graph the data from your table of values that represents this scenario using the Desmos tool below.

Access multimedia content

Compare your answer:

A graph with four labeled points: (3, 5), (2, 1), (negative 2, negative 5) and (0, negative 6).

7.

Now, in the Desmos window graph $y = x$ .

Compare your answer:

A graph with a line passing through four labeled points: (3, 5), (2, 1), (negative 2, negative 5) and (0, negative 6).

8.

How does the line need to move to capture all of the points?

Compare your answer:

Answers may vary.

Increase the y-intercept or the slope to capture the point $(4, 5)$ .

9.

Using desmos, adjust the slope and/or the $y$ -intercept to find an inequality that could be a possibility.

Compare your answer:

Answers may vary.

These are possible representation of where the line could on the graph

Less than $y < 2 x$

Graph of an inequality on the coordinate plane. The region is shaded red below and to the right of the dashed boundary line. Four points lie in the solution region.

Less than or equal to $y^{2} x - 1$

Graph of an inequality on the coordinate plane. The region is shaded red below and to the right of the solid boundary line. Four points lie in the solution region.

Greater than $y > x - 7$

Graph of an inequality on the coordinate plane. The region is shaded red above and to the left of the dashed boundary line. Four points lie in the solution region.

Greater than or equal to $y > x - 6$

Graph of an inequality on the coordinate plane. The region is shaded red above and to the left of the solid boundary line. Four points lie in the solution region.

Not a solution

Graph of an inequality on the coordinate plane. The region is shaded red below and to the right of the dashed boundary line. Only three points lie in the solution region. One point does not satisfy the inequality.

10.

Use the graph from number 7. Shade the correct side of the boundary line to represent the inequality $y > 3 x - 2$ .

Compare your answer:

Graph of an inequalilty on the coordinate plane.

Self Check

Which table of values can be used to graph the boundary line of the inequality $y > - 3 x + 1$ ?

Additional Resources

Connecting Representations of Inequalities

Tables can be used to help you to write and solve inequalities in two variables.

Part 1: Writing Inequalities in Two Variables Given A Set of Data in a Table

Write a linear inequality in two variables based on the given data:

$x$	$y$
-4	-3
-2	-2
0	-1
2	0

Remember that for an inequality the points could be on the line, above the the line, or below the line. The type of inequality will differ based on these conditions.

On the line	$\leq$ or $\geq$
Above the line	$>$
Below the line	$<$

If you are given a set of possible points like the table above, there could be multiple lines that the data points could represent. Let’s figure out a few possibilities for this given set.

Step 1 - Graph the points on a coordinate plane.

A coordinate plane with five labeled points: (5, 8), (3, 5), (1, 1), (3, -5), and (-4, -6). Each point is marked with a green dot and labeled with its coordinates.

Step 2 - Determine possible lines.

Remember that the dots can be on the line, but they all have to be either above or below to create an inequality. If the line has points on both sides, then the line isn’t a possibility.

Let’s try a few possibilities. We can start with the equation $y = x$ to see if the points are above the line or below the line.

A graph of a red line that has a positive slope passes through points (0,0) and (1, 1). Additional green points are at (3,5), (5, 8), (negative 4, negative 6) and (3, negative 5). All points are labeled with their coordinates.

The equation $y = x$ cannot be a possibility because there are points above and below the line.

Step 3 - Adjust the slope or $y$ -intercept

We can change the slope and the y-intercept to see how we can shift the line to be able to capture all the points. Let’s move the line to below the $(3, - 5)$ and $(- 4, - 6)$ . We can leave the slope as 1 and change the y-intercept to -7.

A graph of a red line with positive slope passing through it. Labeled points are at (3, 5), (5, 8), (1, 1), (-4, -6), and (3, -5) marked with green dots. the line does not intersect any of the labeled points.

There is still one point below the line. We could decrease the slope of the line to $m = 13$ .

A graph showing five green points labeled (3, 5), (5, 8), (1, 1), (3, -5), and (-4, -6), with a red line that has a positive slope. The line does not intersect any of the labeled points.

Now all of the points are above the line. This means that the value of $y$ is always greater than the $x$ value.

So, one of the possible solutions could be $y > 13 x - 7$ .

Part 2: Using Tables to Solve Inequalities in Two Variables

Graph the linear inequality $y < \frac{1}{2} x - 1$ using a table.

Step 1 - Create a table of values using the related equation $y = \frac{1}{2} x - 1$ to identify points on the boundary line of the inequality.

$x$	$y$
−4	−3
−2	−2
0	-1
2	0

Step 2 - Plot the points and sketch the boundary line.

If the inequality is $\leq$ or $\geq$ , the boundary line is solid.

If the inequality is $<$ or $>$ , the boundary line is dashed.

The inequality sign is $<$ , so we draw a dashed line.

A graph with a dashed teal line passing through points (-4, -3), (-2, -2), (0, -1), (2, 0), and (4, 1) on a grid. The line has a positive slope and crosses the y-axis at -1.

Step 3 - Test a point that is not on the boundary line. Is it a solution of the inequality?

Test $(0, 0)$ as it is not on the boundary line of the inequality.

$y \overset{?}{<} \frac{1}{2} x - 1$

$(0) < \frac{1}{2} (0) - 1$

$0 < - 1$ is False.

So, $(0, 0)$ is not a solution.

Step 4 - Shade in one side of the boundary line.

If the test point is a solution, shade in the side that includes the point.

If the test point is not a solution, shade in the opposite side.

The test point $(0, 0)$ is not a solution, so shade in the side of the boundary line that does not include $(0, 0)$ .

A graph with a shaded region below the dashed line representing y equals one-half times x minus 1.

All points in the shaded region and not on the boundary line represent the solutions to $y < \frac{1}{2} x - 1$ .

Try it

Try It: Graph Linear Inequalities in Two Variables

1.

Is the inequality $y < 3 x + 6$ a solution for the table of values? Explain your reasoning.

$x$	$y$
-4	-3
-2	-2
0	-1
2	0

Compare your answer:

No, the point $(- 4, - 6)$ sits on the line so it can’t be greater than or less than.

2.

Use a table to graph the inequality $y = 2 x + 3$ .

Compare your answer:

Here’s how to solve an inequality using a table:

Step 1 - Create a table of values using the related equation $y = 2 x + 3$ to identify points on the boundary line of the inequality.

x	y
−4	−5
−2	−1
0	3
2	7

Step 2 - Plot the points and sketch the boundary line.

If the inequality is $\leq$ or $\geq$ , the boundary line is solid.

If the inequality is $<$ or $>$ , the boundary line is dashed.

The inequality sign is , so we draw a solid line.

A graph with a dashed teal line passing through points (-4, -5), (-2, -1), (0, 3), and (2, 7) on a grid. The line has a positive slope and crosses the y-axis at 3.

Step 3 - Test a point that is not on the boundary line. Is it a solution of the inequality?

Test $(0, 0)$ as it is not on the boundary line of the inequality.

$y \leq 2 x + 3$

$(0) \overset{?}{\leq} 2 (0) + 3$

$0 \leq 3$ is True.

So, (0, 0) is a solution.

Step 4 - Shade in one side of the boundary line.

If the test point is a solution, shade in the side that includes the point.

If the test point is not a solution, shade in the opposite side.

The test point $(0, 0)$ is a solution, so shade in the side of the boundary line that includes $(0, 0)$ .

A graph with a shaded region below the dashed line representing y equals two times x plus 3.

All points in the shaded region and not on the boundary line represent the solutions to $y \leq 2 x + 3$ .

2.13.4 Solving Inequality Problems Using Tables

Activity

Additional Resources

Connecting Representations of Inequalities

Try It: Graph Linear Inequalities in Two Variables