Inequality Constraints

Solve the inequality: 2x + 5 > 11

• x > 3

• x < 3

• x > 5

• x = 3

Solve the inequality: -3x + 3 ≤ 15

• x ≤ 4

• x ≥ 4

• x ≤ -4

• x ≥ -4

Emma is planning a birthday party and wants balloons and party hats. She has $20 to spend. Each balloon costs $1, and each party hat costs $2. She wants to buy at least 5 balloons. She needs to have enough items so that each of her 12 friends gets either a balloon or a hat. Emma doesn't want to buy more than 15 items in total. Let x represent the number of balloons and y represent the number of party hats. Select all the inequalities represented by the situation.

• 1x + 2y < 20

• x > 5

Answer explanation

The total cost of balloons and hats must be less than or equal to $20, represented by 1x + 2y ≤ 20. Emma wants at least 5 balloons, so x ≥ 5. The other options do not accurately reflect the constraints.

Katie wants to have at least $200 in her bank account by the end of the month. She currently has $25 in her bank account. Katie makes $12 for each dog that she walks. Which inequality could help you find how many dogs Katie would have to walk to reach her goal?

• 25 - 12x ≥ 200

• 25 + 12x ≥ 200

• 13x ≥ 200

• 25 + 12x < 200

Answer explanation

To find how many dogs Katie needs to walk, we start with her current amount ($25) and add the money she earns from walking dogs ($12 per dog). The inequality 25 + 12x ≥ 200 represents this situation.

Which inequality represents "less than 4 ounces"?

• x > 4

• x < 4

Answer explanation

The phrase "less than 4 ounces" indicates that the value of x must be below 4. Therefore, the correct inequality is x < 4, which accurately represents this condition.

A student takes a test with multiple choice (m) and short answer (s) problems. Multiple choice problems are worth 2 points, short answers are worth 5 points, there are over 50 questions on the test, and no more than 200 points possible. Identify all the constraints that restrict the problem.

• m + s > 50

• 2m + 5s > 50

• m > 0

Answer explanation

The constraints include: m + s > 50 (more than 50 questions), 2m + 5s ≤ 200 (total points limit), and m > 0 (at least one multiple choice question). The other options do not correctly represent the problem's restrictions.

A class is planning a garden. They will plant tomatoes and carrots and have space for no more than 20 plants. They must plant at least 5 tomato plants and need to plant more carrot plants than tomato plants. Their plant budget is $30 with each tomato plant costing $2 and each carrot plant costing $1.

Let x represent the number of tomato plants and y represent the number of carrot plants. Choose all the inequalities represented by the situation.

• x + y < 20

• x + y > 20

Answer explanation

The class can plant a maximum of 20 plants, so the correct inequality is x + y ≤ 20. This ensures they do not exceed their planting space.

A limo driver needs to make more than $450 in one day. He charges a rental fee of $375 plus $0.85 per mile. What inequality represents the number of miles, x, he would have to drive to reach his goal?

• 0.85x + 375 > 450

• 0.85x + 375 < 450

• 0.85 + 375x > 450

• 0.85 + 375x < 450

Answer explanation

To find the number of miles, x, the driver needs to drive, we set up the inequality: rental fee ($375) plus the cost per mile ($0.85x) must be greater than $450. Thus, the correct inequality is 0.85x + 375 > 450.

You are planning to buy lunch for yourself and a friend. You need to purchase at least 2 hamburgers and 2 milkshakes. Given this situation, what constraints are on the variables involved?

• Hamburgers and fries must each be greater than 0.

• Hamburgers and fries must each be greater than or equal to 0.

• Hamburgers and fries must each be less than or equal to 2.

• Hamburgers and fries must each be greater than or equal to 2.

The grocery store is selling tissues for $1 per box and paper towels for $3 per roll. The store spent a total of $400 on inventory. To make a profit, they need to make more money from sales than the $400 they spent. What linear inequality and constraints best describe the situation above?