Graphing Solutions and Interpreting Points

2.12.3 Graphing Solutions and Interpreting Points

2.12.3 • Graphing Solutions and Interpreting Points

Activity

For questions 1 – 7, use the following scenario:

A vendor at the Saturday Market makes $9 profit on each necklace she sells and $5 profit on each bracelet. Find a combination of necklaces and bracelets that she could sell to make the profit listed in each question.

Exactly $100 profit.

Compare your answer: Your answer may vary, but here is a sample. 10 necklaces and 2 bracelets.

More than $100 profit.

Compare your answer: Your answer may vary, but here is a sample. 0 necklaces and 21 bracelets.

Write an equation whose solution is the combination of necklaces $(n)$ and bracelets $(b)$ she could sell and make exactly $100 profit.

Compare your answer: Your answer may vary, but here is a sample. $9 n + 5 b = 100$ , where $n$ represents the number of necklaces sold and $b$ represents the number of bracelets.

Write an inequality whose solutions are the combinations of necklaces $(n)$ and bracelets $(b)$ she could sell and make more than $100 profit.

Compare your answer: Your answer may vary, but here is a sample. $9 n + 5 b > 100$ where $n$ represents the number of necklaces sold and $b$ represents the number of bracelets.

Use the graphing tool or technology outside the course.Graph the linear inequality $y > \frac{5}{2} x - 4$ using the Desmos tool below.

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Compare your answer:

The graph of an inequality in the first quadrant. The x-axis represents the number of necklaces and the y-axis represents the number of bracelets.

Is (3,18.6) a solution to the inequality?

Both.

Explain your reasoning.

Compare your answer: Your answer may vary, but here is a sample.
The answer is both because:
Yes, it is a solution. The point (3,18.6) is in the shaded region.
No, it is not a solution. The vendor can't sell 18.6 bracelets.

Are you ready for more?

Extending Your Thinking

Write an inequality using two variables $x$ and $y$ where the solution would be represented by shading the entire coordinate plane.

$x^{2} + y^{2} \geq 0$

Write an inequality using two variables $x$ and $y$ where the solution would be represented by not shading any of the coordinate plane.

$x^{2} + y^{2} < 0$

Self Check

An artist is designing a sculpture to be displayed hanging in the lobby of a museum. She is using $x$ wooden rods and $y$ metal pipes.

Each wooden rod, $x$ , weighs 4 pounds.
Each metal pipe, $y$ , weighs 7 pounds.
The total weight of these materials cannot be more than 510 pounds.

Which of these ordered pairs represents an amount of materials that meets the artist’s constraints?

$(70, 20)$
$(50, 60)$
$(20, 70)$
$(60, 50)$

Additional Resources

Graph Linear Inequalities in Two Variables

Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.

Example

How to Graph a Linear Equation in Two Variables

Graph the linear inequality $y \geq \frac{3}{4} x - 2$ .

Step 1 - Identify and graph the boundary line.

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

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

Replace the inequality sign with an equal sign to find the boundary line

$y = \frac{3}{4} x - 2 .$

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

A graph on the coordinate plane of a line that passes through (0, -2) and (4, 1).

Step 2 - Test a point that is not on the boundary line. Is it a solution of the inequality?

We’ll test $(0, 0)$ .

Is it a solution of the inequality?

At $(0, 0)$ , is $y \geq \frac{3}{4} x - 2 .$ ?

$0 \overset{?}{\geq} \frac{3}{4} (0) - 2$

$0 \geq - 2$

So, $(90, 0)$ is a solution.

Step 3 - 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 to $y \geq \frac{3}{4} x - 2 .$ So we shade in that side.

A graph with a shaded region above the boundary line representing y equals three-fourths times x minus two and covering the upper left side of the coordinate plane. The boundary line of the inequality is solid.

All points in the shaded region and on the boundary line represent the solutions to $y \geq \frac{3}{4} x - 2 .$

Try it

Graph Linear Inequalities in Two Variables

Graph the linear inequality $y > \frac{5}{2} x - 4$ . Use the graphing tool or technology outside the course. Graph the inequality using the Desmos tool below.

Compare your answer:

All points in the shaded region represent the solutions to $y > \frac{5}{2} x - 4$

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