Chapter 7

Simplify. (Assume all denominators are nonzero.) $$ 5 x-5+20-9 x 2 x 2-15 x+25 $$

Joe can assemble a computer by himself in 1 hour. Working with an assistant, he can assemble a computer in 40 minutes. How long would it take his assistant to assemble a computer working alone?

Construct a mathematical model given the following. \(y\) varies directly as the square of \(x\) and inversely as \(z\) and the square of \(w,\) where \(y=14\) when \(x=4, w=2,\) and \(z=2\).

Simplify. (Assume all denominators are nonzero.) $$ x x-5-2 x-3-5(x-3) \times 2-8 x+15 $$

The teacher's assistant can grade class homework assignments by herself in 1 hour. If the teacher helps, then the grading can be completed in 20 minutes. How long would it take the teacher to grade the papers working alone?

Simplify. $$ x x+3+1 x-3-15-x(x+3)(x-3) $$

Simplify. (Assume all denominators are nonzero.) $$ 2-52 x-3 x 24 x+3 $$

Construct a mathematical model given the following. \(y\) varies directly as the square root of \(x\) and inversely as \(z\) and the square of \(w,\) where \(y=27\) when \(x=9, w=1 / 2,\) and \(z=4\).

State the restrictions and then simplify. $$ 16 x 2-1(4 x+1) 2 $$

Simplify. (Assume all denominators are nonzero.) $$ 3 x 2 x-1-x-4 x+4+12(2-x) 2 x 2+7 x-4 $$

Solve. $$4 x-7 x-5=3 x-2 x-5$$

A larger pipe fills a water tank twice as fast as a smaller pipe. When both pipes are used, they fill the tank in 5 hours. If the larger pipe is left off, then how long would it take the smaller pipe to fill the tank?

Simplify. $$ 2 x 3 x-1-13 x+1+2(x-1)(3 x-1)(3 x+1) $$

Applications involving variation. Revenue in dollars is directly proportional to the number of branded sweat shirts sold. If the revenue earned from selling 25 sweat shirts is \(\$ 318.75,\) then determine the revenue if 30 sweat shirts are sold.

Simplify. (Assume all denominators are nonzero.) $$ 1 \times 2+8 x-9-1 x 2+11 x+18 $$

Solve. $$x x 2-9=1 x-3$$

A newer printer can print twice as fast as an older printer. If both printers working together can print a batch of flyers in 45 minutes, then how long would it take the newer printer to print the batch working alone?

Simplify. $$ 4 x 2 x+1-x x-5+16 x-3(2 x+1)(x-5) $$

Applications involving variation. The sales tax on the purchase of a new car varies directly as the price of the car. If an \(\$ 18,000\) new car is purchased, then the sales tax is \(\$ 1,350\). How much sales tax is charged if the new car is priced at \(\$ 22,000 ?\)

Simplify. (Assume all denominators are nonzero.) $$ 4 x 2+13 x+36+3 x 2+6 x-27 $$

Divide. (Assume all denominators are nonzero.) $$ y 2-7 y+10 y 2+5 y-14 \quad 2 y 2-9 y-5 y 2+14 y+49 $$

Solve. $$3 x+4 x-8-28-x=1$$

Working alone, Henry takes 9 hours longer than Mary to clean the carpets in the entire office. Working together, they clean the carpets in 6 hours. How long would it take Mary to clean the office carpets if Henry were not there to help?

Simplify. $$ x 3 x+2 x-2+43 x(x-2) $$

Simplify. (Assume all denominators are nonzero.) $$ 1 x+5+4 x-22 x-2-1 x+5 $$

Applications involving variation. The price of a share of common stock in a company is directly proportional to the earnings per share (EPS) of the previous 12 months. If the price of a share of common stock in a company is \(\$ 22.55\) and the EPS is published to be \(\$ 1.10,\) then determine the value of the stock if the EPS increases by \(\$ 0.20\).

Working alone, Monique takes 4 hours longer than Audrey to record the inventory of the entire shop. Working together, they take inventory in 1.5 hours. How long would it take Audrey to record the inventory working alone?

Simplify. $$ -2 x x+6-3 x 6-x-18(x-2)(x+6)(x-6) $$

Simplify. (Assume all denominators are nonzero.) $$ 3 x-1-2 x+32 x+3+1 x-3 $$

Applications involving variation. The distance traveled on a road trip varies directly with the time spent on the road. If a 126-mile trip can be made in 3 hours, then what distance can be traveled in 4 hours?

Divide. (Assume all denominators are nonzero.) $$ x 2-7 x-18 x 2+8 x+12 \div x 2-81 x 2+12 x+36 $$

Solve. $$3 x=1 x+1+13 x(x+1)$$

Jerry can lay a tile floor in 3 hours less time than Jake. If they work together, the floor takes 2 hours. How long would it take Jerry to lay the floor by himself?

Simplify. (Assume all denominators are nonzero.) $$ x x+1-2 x+3 x 3 x+4+1 x+1 $$

Applications involving variation. The circumference of a circle is directly proportional to its radius. If the circumference of a circle with radius 7 centimeters is measured as \(14 \pi\) centimeters. then find the constant of proportionalitv.

Solve. $$x x-1-34 x-1=9 x(4 x-1)(x-1)$$

Jeremy can build a model airplane in 5 hours less time than his brother. Working together, they need 6 hours to build the plane. How Iong would it take Jeremy to build the model airplane working alone?

Simplify. $$ x x 2-2 x-3+2 x-3 $$

Applications involving variation. The area of circle varies directly as the square of its radius. If the area of a circle with radius 7 centimeters is determined to be \(49 \pi\) square centimeters, then find the constant of proportionality.

Given \(f(x)=x+13 x\) and \(g(x)=2 x-8\), calculate \((f-g)(x)\) and state the restrictions.

Solve. $$1 x-4+x x-2=2 x 2-6 x+8$$

Harry can paint a shed by himself in 6 hours. Jeremy can paint the same shed by himself in 8 hours. How long will it take them to paint two sheds working together?

Simplify. $$ 1 x+5-x 2 x 2-25 $$

Applications involving variation. The surface area of a sphere varies directly as the square of its radius. When the radius of a sphere measures 2 meters, the surface area measures \(16 \pi\) square meters. Find the surface area of a sphere with radius 3 meters.

Divide. (Assume all denominators are nonzero.) $$ 5 y 2(y-3) 4 x 3 \div 25 y(3-y) 2 x 2 $$

Joe assembles a computer by himself in 1 hour. Working with an assistant, he can assemble 10 computers in 6 hours. How long would it take his assistant to assemble 1 computer working alone?

Applications involving variation. The volume of a sphere varies directly as the cube of its radius. When the radius of a sphere measures 3 meters, the volume is \(36 \pi\) cubic meters. Find the volume of a sphere with radius 1 meter.

Jerry can lay a tile floor in 3 hours, and his assistant can do the same job in 4 hours. If Jerry starts the job and his assistant joins him 1 hour later, then how long will it take to lay the floor?

Applications involving variation. With a fixed height, the volume of a cone is directly proportional to the square of the radius at the base. When the radius at the base measures 10 centimeters, the volume is 200 cubic centimeters. Determine the volume of the cone if the radius of the base is halved.

Simplify. $$ 16+1 \times 136-1 \times 2 $$