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More boring questions for school algebra prep … sorry

Due: Jul 25th

Question

  1. Simplify the following expression: 3(2x^2 - 4x) - 5(3x^2 + 2x) and express in standard form.
  2. Solve the equation 5(3x - 4) - 2(4x - 5) = 0 for x.
  3. Solve the inequality: 3(2x - 5) + 4 < 7(3x - 2).
  4. The ratio of two numbers is 3:5 and their sum is 320. What are the two numbers?
  5. A farmer has both chickens and sheep on his farm. He counts 40 heads and 112 legs. How many chickens and how many sheep does he have?
  6. You have 2000ml of a 25% saline solution. How much saline should you add to make it a 40% saline solution?
  7. If a runner’s speed is 7.5 miles/hour, how far will she run in 45 minutes at this speed?
  8. A car rental agency offers two plans: Plan A charges $50/day and $0.20/mile. Plan B charges $30/day and $0.40/mile. For a one-day rental, how many miles would you need to drive for Plan B to be cheaper?
  9. An investment account starts with $5000 and earns 2% interest compounded annually. Write an equation for the balance, B, after n years.
  10. Solve for x: (x^2 - 9) / (x - 3) = x + 3.
  11. Calculate the area of a circle with a radius of 7 inches to the nearest square inch. Use pi = 3.14.
  12. Calculate the surface area of a cylinder with a radius of 3 inches and a height of 5 inches to the nearest square inch. Use pi = 3.14.

Solutions

  1. Simplifying the expression: 3(2x^2 - 4x) - 5(3x^2 + 2x) yields -9x^2 - 2x.
  2. Solving the equation 5(3x - 4) - 2(4x - 5) = 0 gives x = 1.
  3. Solving the inequality: 3(2x - 5) + 4 < 7(3x - 2) gives x > 6/5.
  4. For the ratio problem, let’s denote the two numbers as 3x and 5x. Their sum is 320, so 3x + 5x = 320. Solving for x gives x = 40. Therefore, the numbers are 120 and 200.
  5. Let’s denote the number of chickens as c and sheep as s. We have two equations: c + s = 40 (each animal has one head) and 2c + 4s = 112 (chickens have 2 legs and sheep have 4). Solving these gives c = 24 and s = 16.
  6. Let’s denote the amount of saline to add as x (in ml). The equation is 2000 * 0.25 + x = (2000 + x) * 0.40. Solving for x gives x = 1000 ml.
  7. 7.5 miles/hour is equivalent to 7.5 * (45 / 60) = 5.625 miles in 45 minutes.
  8. Plan A costs $50 + $0.20m and Plan B costs $30 + $0.40m, where m is the miles driven. Equating the two costs and solving for m gives m = 100 miles.
  9. The equation for the balance B after n years would be B = 5000 * (1 + 0.02)^n.
  10. Simplifying (x^2 - 9) / (x - 3) = x + 3 gives x = 6.
  11. The area of a circle with radius 7 inches is A = pi * r^2 = 3.14 * 7^2 = 153.94 square inches, rounded to the nearest square inch gives 154 square inches.
  12. The surface area of a cylinder is given by A = 2 * pi * r * (r + h). Substituting the given values gives A = 2 * 3.14 * 3 * (3 + 5) = 150.72 square inches. Rounded to the nearest square inch gives 151 square inches.
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