Writing an Inequality to Represent a Constraint

2.12.2 Writing an Inequality to Represent a Constraint

2.12.2 • Writing an Inequality to Represent a Constraint

Activity

For questions 1 – 8, use the following scenario:

The homeowner is worried about the work needed to maintain a grass lawn and flower beds, so she is now looking at some low-maintenance materials.

She is considering a combination of the materials shown to cover the yard. Her budget is still $3,000.

artificial turf: $15 per square foot

gravel: $3 per square foot

Here is a graph representing some constraints in this situation.

The graph shows a line going through $(500, 100)$ . In this situation, what does the point $(500, 100)$ mean?

Compare your answer:

The cost of installing 500 square feet of gravel and 100 square feet of artificial turf is $3000.

Write an equation that the line represents.

Compare your answer:

$3 x + 15 y = 3000$

What do the solutions to the equation mean?

Compare your answer:

The solutions are all combinations of square feet of gravel and artificial turf that would cost $3000 to install.

The point $(600, 200)$ is located to the right and above the line. Does that combination of turf and gravel meet the homeowner's constraints? Be prepared to show your reasoning.

Compare your answer:

No. Answers may vary, but here is a sample. If she installs 600 square feet of gravel and 200 square feet of turf, the cost is $3 (600) + 15 (200)$ , which is $1800 + 3, 000$ or $$ 4800$ .

Choose another point in the same region (to the right and above the line). Check if the combination meets the homeowner's constraints.

Compare your answer:

Answers may vary, but here is a sample. $(900, 50)$ . This combination would cost more than $3000 to install.

The point $(200, 100)$ is located to the left and below the line. Does that combination of turf and gravel meet the homeowner's constraints? Be prepared to show your reasoning.

Compare your answer:

Yes. Answers may vary, but here is a sample: $3 (200) + 15 (100) = 2100$ . This amount is less than $3000.

Choose another point in the same region (to the left and below the line). Check if the combination meets the homeowner's constraints.

Compare your answer:

Answers may vary, but here is a sample: $(600, 25)$ . This combination would cost under $3000 to install.

Write an inequality that represents the constraints in this situation. Explain what the solutions mean and show the solution region on the graph.

Compare your answer:

Answers may vary, but here is a sample:

$3 x + 15 y \leq 3000$ . The solutions represent all possible combinations of square feet of gravel and turf that would cost no more than $3000 to install.

The graph of an inequality in the first quadrant of the coordinate plane. The x-axis label indicates it represents the squre feet of gravel sod. The y-axis label says it represents the square feet of artificial turf.

Video: Writing an Inequality to Represent a Constraint

Watch the following video to learn more about representing constraints using inequalities.

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Self Check

A builder is using a mix of different tiles to remodel a kitchen. The marble tiles cost,

x

, $15 per square and the ceramic tiles,

y

, cost $7 per square. The total amount allowed for tiles is at most $900. Which inequality represents this situation?

$22(x\;+\;y)\;\leq\;900$
$15x - 7y < 900$
$15x + 7y < 900$
$15x\;+\;7y\;\leq\;900$

Additional Resources

Writing Inequalities to Represent Constraints

Example

Elena needs to earn at least $450 a week during her summer break to pay for college. She works two jobs. One as a swimming instructor that pays $9 an hour and the other as an intern in a genetics lab for $22.50 per hour. How many hours does Elena need to work at each job to earn at least $450 per week?

Let $x$ be the number of hours she works teaching swimming and let $y$ be the number of hours she works as an intern.

Write an inequality that would model this situation.

She earns $9 per hour at the job as a swimming instructor and $22.50 in the genetics lab. At each job the number of hours multiplied by the hourly wage will give the amount earned at that job.

Mathematical expression: 9 times x plus 22.5 times y is greater than or equal to 450. Above the inequality, labels indicate the term representing 9 times x is the amount earned as a swimming instructor, and the term representing 22.5 times y is the amount earned in the genetics lab. The phrase "is at least 450 dollars" represents the label for greater than or equal to 450.

Try it

Writing Inequalities to Represent Constraints

A baker sells pies and cakes at the bakery. Each pie sells for $14 and each cake sells for $23. Between all the sales, the baker wants to earn at least $525. How many pies and cakes does the baker need to sell to meet their goal?

Let $p$ be the number of pies sold and let $c$ be the number of cakes sold. Write an inequality that would model this situation.

Compare your answer: You answer may vary, but here is a sample.

$14 p + 23 c \geq 525$