Exploring Real-World Inequalities: Graphs & Applications

A farmer wants to create a rectangular garden with an area of at least 200 square meters. If the length of the garden is represented by x meters, what inequality represents the possible values for the width, y, if the width must be at least 5 meters?

• y < 200/x

• y <= 5

• y >= max(200/x, 5)

• y > 200/x

A company produces a product that has a profit represented by the equation P(x) = -2x^2 + 40x - 100, where x is the number of units sold. Determine the range of units that must be sold to ensure a profit greater than zero. Graph the inequality and interpret the solution.

• 4 to 7

• 10 to 12

• 1 to 5

• The range of units that must be sold to ensure a profit greater than zero is 6 to 9.

A car rental company charges a flat fee of $50 plus $0.20 per mile driven. If a customer wants to keep their total cost under $100, what inequality represents the maximum number of miles they can drive?

• m < 150

• m < 300

• m < 250

• m < 200

A basketball player can make a maximum of 80% of their free throws. If they attempt x free throws, what inequality represents the number of successful free throws they must make to ensure they score at least 20 points?

• s <= 20

• s >= 20

• s < 20

• s > 20

A rectangular swimming pool has a length that is 3 meters longer than its width. If the area of the pool must be less than 150 square meters, write an inequality to represent the relationship between the length and width of the pool. What are the possible dimensions?

• Width (w) in (0, 15) meters; Length (l) in (3, 18) meters.

• Possible dimensions: Width (w) in (0, 12) meters; Length (l) in (3, 15) meters.

• Width (w) in (0, 10) meters; Length (l) in (3, 13) meters.

• Width (w) in (0, 5) meters; Length (l) in (3, 8) meters.

A local concert venue has a seating capacity of 500. If tickets are sold for $20 each, and the venue wants to ensure that at least $8,000 is made from ticket sales, what inequality represents the minimum number of tickets that must be sold?

• x ≥ 400

• x ≥ 450

• x ≤ 400

• x < 400

A student is saving money for a new laptop that costs $800. If they save $50 each week, what inequality can represent the number of weeks, w, they need to save to afford the laptop?

• 50w = 800

• 50w >= 800

• 50w < 800

• 50w <= 800

A rectangular plot of land has a perimeter of at most 100 meters. If the length is represented by x meters, write an inequality for the width, y, in terms of x. What are the possible dimensions of the plot?

• y = 100 - x, where x > 0 and x < 100.

• y ≤ 50 - x, where x > 0 and x < 50.

• y ≤ 25 - x, where x > 0 and x < 25.

• y ≥ 50 - x, where x > 0 and x < 100.

A factory produces widgets at a cost represented by the equation C(x) = 3x^2 - 12x + 50. If the factory wants to keep production costs below $100, what inequality must be satisfied?

• 3x^2 - 12x - 50 < 0

• 3x^2 - 12x + 50 < 100

• 3x^2 - 12x - 50 > 0

• 3x^2 - 12x + 50 > 0

A school is planning a field trip and has a budget of $1,200. If each student pays $30, how many students can attend the trip while staying within budget? Write an inequality to represent this situation.

• x ≤ 20

• x ≤ 30

• x ≤ 40

• x ≤ 50