Solving Systems of Inequalities with Word Problems

Front

Luka and Dana are making cookies for the Westland Pantry. They only have up to 6 hours to bake! They can bake 32 chocolate chip cookies in an hour (x), and 20 snickerdoodle cookies in an hour (y). They need to make at least 150 cookies. Which inequalities represent the situation?

Back

x + y ≤ 6; 32x + 20y ≥ 150

Answer explanation

The correct inequalities are x + y ≤ 6 (time constraint) and 32x + 20y ≥ 150 (cookie requirement). This ensures they do not exceed 6 hours while making at least 150 cookies.

Front

Kate is selling bracelets (x) for $2 and earrings (y) for $3. She needs to make at least $100 and can make up to 40 pieces of jewelry. Which inequalities represent this situation?

Back

2x + 3y ≥ 100, x + y ≤ 40

Answer explanation

The correct inequalities are 2x + 3y > 100 (total earnings) and x + y ≤ 40 (total pieces). This ensures Kate makes at least $100 while not exceeding 40 items.

Front

What's the difference between systems of equations and systems of inequalities?

Back

Systems of equations have 1 solution, no solution, or infinite solutions while systems of inequalities can have a range of solutions.

Answer explanation

The correct choice highlights that systems of equations can have a single solution, no solution, or infinitely many solutions, while systems of inequalities represent a range of solutions, often depicted as shaded regions on a graph.

Front

Cole's basketball challenge: If x is the number of 2-point shots, and y is the number of 3-point shots, what inequalities represent scoring at least 20 points in less than 10 shots?

Back

x + y < 10, 2x + 3y ≥ 20

Answer explanation

Cole needs to score at least 20 points, represented by the inequality 2x + 3y ≥ 20. He must also take fewer than 10 shots, represented by x + y < 10. Thus, the correct inequalities are x + y < 10 and 2x + 3y ≥ 20.

Front

Write the system of inequalities for a theater with 300 seats, charging $7.50 for adults and $5.50 for children, expecting to make at least $2000 per showing.

Back

x+y≤300
7.5x+5.5y≥2000

Answer explanation

The first inequality, x+y≤300, ensures the total number of tickets sold does not exceed the theater's seating capacity. The second inequality, 7.5x+5.5y≥2000, guarantees that the revenue from ticket sales meets the minimum expected amount.

Front

JoAnn's work hours and earnings constraints: JoAnn likes her job as a baby-sitter, but it pays only $3 per hour. She has been offered a job as a tutor that pays $6 per hour. Because of school, her parents only allow her to work a maximum of 15 hours per week. How many hours can JoAnn tutor and baby-sit and still make at least $66 per week?

Back

b + t < 15
3b + 6t > 66

Answer explanation

JoAnn can work a maximum of 15 hours (b + t < 15) and needs to earn at least $66 (3b + 6t > 66). The correct choice reflects these conditions, ensuring she balances her hours and earnings.

Front

System of equations for David's concession stand: Nachos cost $1.50 each, sodas cost $0.50 each. Total sales: $78.50. Total items sold: 87.

Back

1.5x+0.5y=78.5, x+y=87

Answer explanation

The first equation, 1.5x + 0.5y = 78.5, represents the total revenue from nachos (x) and sodas (y). The second equation, x + y = 87, represents the total number of items sold. Thus, the correct system is: 1.5x + 0.5y = 78.5 and x + y = 87.

Front

Which point is a solution? Options: (0, 6), (-3, 4), (1, 0), (-4, 3)

Back

(1, 0)

Answer explanation

The point (1, 0) is a solution because it satisfies the equation or condition given in the problem. The other points do not meet the criteria, making (1, 0) the correct choice.

Front

Write the inequality

Back

y≥-5x-3

Answer explanation

The correct inequality is y≥-5x-3 because it indicates that y can be equal to or greater than the line defined by -5x-3, which includes the boundary line itself.

Front

Luka and Dana need to bake at least 150 cookies in up to 6 hours. They can bake 32 chocolate chip cookies in an hour (x), and 20 snickerdoodle cookies in an hour (y). Which one of these is a valid solution? Options: (3, 2), (1, 4), (2, 8), (5, 1)

Back

(5, 1)

Answer explanation

To find a valid solution, we check the time and cookie count. (5, 1) means 5 hours baking chocolate chip (160 cookies) and 1 hour baking snickerdoodles (20 cookies), totaling 180 cookies, which meets the requirement.