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Algebra 1

2.12.2 Writing an Inequality to Represent a Constraint

Algebra 12.12.2 Writing an Inequality to Represent a Constraint
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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.
insert alt text
1.

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

2.

Write an equation that the line represents.

3.

What do the solutions to the equation mean?

4.

The point (600,200)(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.

5.

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

6.

The point (200,100)(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.

7.

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

8.

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

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?
  1. 22 ( x + y ) ≤ 900
  2. 15 x − 7 y < 900
  3. 15 x + 7 y < 900
  4. 15 x + 7 y ≤ 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

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

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