Chapter 7

Translate the following sentences into a mathematical formula. The period, \(T\), of a pendulum is directly proportional to the square root of its length, \(L\).

Solve. $$5 x+5=3 x+1$$

Simplify. $$ 2 x x-1-3 x+4 x-1+x-2 x-1 $$

The difference of the reciprocals of two consecutive positive odd integers is \(2 / 99 .\) Find the integers.

Simplify. (Assume all denominators are nonzero.) $$ 2-121+34 $$

Translate the following sentences into a mathematical formula. The time, \(t\), it takes an object to fall is directly proportional to the square root of the distance, \(d\), it falls.

State the restrictions and simplify. $$ 3 x 2+9 x 2 x 3-18 x $$

Solve. $$x 6-6 x=0$$

If 3 times the reciprocal of the larger of two consecutive integers is subtracted from 2 times the reciprocal of the smaller, then the result is \(1 / 2\). Find the two integers.

Construct a mathematical model given the following. \(y\) varies directly as \(x,\) and \(y=30\) when \(x=6\).

Simplify. $$ -3 y+25 y-10+y+75 y-10-3 y+45 y-10 $$

If 3 times the reciprocal of the smaller of two consecutive integers is subtracted from 7 times the reciprocal of the larger, then the result is 1/2. Find the two integers.

Construct a mathematical model given the following. \(y\) varies directly as \(x,\) and \(y=52\) when \(x=4\).

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

A positive integer is 5 less than another. If the reciprocal of the smaller integer is subtracted from 3 times the reciprocal of the larger, then the result is \(1 / 12\). Find the two integers.

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

Construct a mathematical model given the following. \(y\) is directly proportional to \(x,\) and \(y=12\) when \(x=3\).

Solve. $$2 x x+5=16-x$$

Simplify. $$ 3 x+5(2 x-1)(x-6)-x+6(2 x-1)(x-6) $$

A positive integer is 6 less than another. If the reciprocal of the smaller integer is subtracted from 10 times the reciprocal of the larger, then the result is \(3 / 7\). Find the two integers.

Simplify. (Assume all denominators are nonzero.) $$ 5 a 2 b-115 a 3(b-1) 2 $$

Construct a mathematical model given the following. \(y\) is directly proportional to \(x\), and \(y=120\) when \(x=20\).

State the restrictions and simplify. $$ 8 x 2-10 x-39-4 x 2 $$

James can jog twice as fast as he can walk. He was able to jog the first 9 miles to his grandmother's house, but then he tired and walked the remaining 1.5 miles. If the total trip took 2 hours, then what was his average jogging speed?

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

Construct a mathematical model given the following. \(y\) varies directly as \(x,\) and \(y=14\) when \(x=10\)

For each problem below, approximate the weight of a 120-pound object at the given height above the surface of earth. (1 mile \(=5,280\) feet). 44,350 feet

State the restrictions and simplify. Given \(f(x)=x-3 x 2+9,\) find \(f(-3), f(0),\) and \(f(3)\).

Multiply. (Assume all denominators are nonzero.) $$ x+1 x-3 \cdot 3-x x+5 $$

Solve. $$9 x 3 x-1-4 x=0$$

On a business trip, an executive traveled 720 miles by jet aircraft and then another 80 miles by helicopter. If the jet averaged 3 times the speed of the helicopter and the total trip took 4 hours, then what was the average speed of the jet?

Simplify. (Assume all denominators are nonzero.) $$ 2 x+13-1 x $$

Construct a mathematical model given the following. \(y\) varies directly as \(x,\) and \(y=2\) when \(x=8\).

For each problem below, approximate the weight of a 120-pound object at the given height above the surface of earth. (1 mile \(=5,280\) feet). 90,000 feet

Solve. $$1-2 x=48 \times 2$$

Simplify. $$ x 2+2 x 2+3 x-28+x-22 x 2+3 x-28 $$

Sally was able to drive an average of 20 miles per hour faster in her car after the traffic cleared. She drove 23 miles in traffic before it cleared and then drove another 99 miles. If the total trip took 2 hours, then what was her average speed in traffic?

Simplify. (Assume all denominators are nonzero.) $$ 23 y-46-1 y $$

Construct a mathematical model given the following. \(y\) varies inversely as \(x\), and \(y=5\) when \(x=7\).

If each share of a company stock is priced at \(\$ 22.40,\) then calculate the \(P /\) E ratio given the following values for the earnings per share. \(\$ 1.40\)

Multiply. (Assume all denominators are nonzero.) $$ 9+x 3 x+1 \cdot 3 x+9 $$

Harry traveled 15 miles on the bus and then another 72 miles on a train. If the train was 18 miles per hour faster than the bus and the total trip took 2 hours, then what was the average speed of the train?

Simplify. (Assume all denominators are nonzero.) $$ 5 y-1210-y y 2 $$

Construct a mathematical model given the following. \(y\) varies inversely as \(x,\) and \(y=12\) when \(x=2\).

If each share of a company stock is priced at \(\$ 22.40,\) then calculate the \(P /\) E ratio given the following values for the earnings per share. \(\$ 1.21\)

Multiply. (Assume all denominators are nonzero.) $$ 12 y 2 y 3(2 y-1) \cdot(2 y-1) 3 y $$

Multiply. (Assume all denominators are nonzero.) $$ 12+5 x \cdot 5 x+25 x $$

Solve. $$1+12 x=12 x-2$$

Simplify. $$ 12+13 x $$

A bus averages 6 miles per hour faster than a trolley. If the bus travels 90 miles in the same time it takes the trolley to travel 75 miles, then what is the speed of each?