Chapter 7

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

Solve. $$ x+12(x-2)+x-6 x=1 $$

Solve. $$x+2 x 2-5 x+4+x+2 x 2+x-2=x-1 x 2-2 x-8$$

Simplify. $$ a 4-a+a 2-9 a+18 a 2-13 a+36 $$

Explain why we need to simplify the numerator and denominator to a single algebraic fraction before multiplying by the reciprocal of the divisor.

The breaking distance of an automobile is directly proportional to the square of its speed. The volume of a right circular cylinder varies jointly as the square of its radius and its height. A right circular cylinder with a 3 centimeter radius and a height of 4 centimeters has a volume of \(36 \pi\) cubic centimeters. Find a formula for the volume of a right circular cylinder in terms of its radius and height.

Solve. $$ 5 x+2 x+1-x x+4=4 $$

Simplify. $$ 3 a-12 a 2-8 a+16-a+24-a $$

The breaking distance of an automobile is directly proportional to the square of its speed. The period, \(T\), of a pendulum is directly proportional to the square root of its length, \(L\). If the length of a pendulum is 1 meter, then the period is approximately 2 seconds. Approximate the period of a pendulum that is 0.5 meter in length.

Solve. $$8 x 2 x-3+4 x 2 x 2-7 x+6=1 x-2$$

Simplify. $$ a 2-142 a 2-7 a-4-51+2 a $$

The breaking distance of an automobile is directly proportional to the square of its speed. The time, \(t\), it takes an object to fall is directly proportional to the square root of the distance, \(d\), it falls. An object dropped from 4 feet will take \(1 / 2\) second to hit the ground. How long will it take an object dropped from 16 feet to hit the ground?

$$ -2 x+14 x 3-49 x $$

Solve for the indicated variable. Solve for \(r: t=D r\)

Simplify. $$ 1 x+3-x x 2-6 x+9+3 x 2-9 $$

If two objects with masses 50 kilograms and 100 kilograms are \(1 / 2\) meter apart, then they produce approximately \(1.34 \times 10-6\) newtons (N) of force. Calculate the gravitational constant.

State the restrictions and then simplify. $$ 2 x 2-7 x-41-4 x 2 $$

Calculate \((f \cdot g)(x)\) and determine the restrictions to the domain. $$ f(x)=1 x \text { and } g(x)=1 x-1 $$

Solve for the indicated variable. Solve for \(b: h=2 A b .\)

State the restrictions and then simplify. $$ 9 x 2-44 x-6 x 2 $$

Solve. $$ 2 x x+5-12 x-3=4-7 x 2 \times 2+7 x-15 $$

Solve for the indicated variable. Solve for \(P: t=I P r\).

Calculate the force in newtons between earth and the moon, given that the mass of the moon is approximately \(7.3 \times 1022\) kilograms, the mass of earth is approximately \(6.0 \times 1024\) kilograms, and the distance between them is on average \(1.5 \times 1011\) meters.

State the restrictions and then simplify. $$ 2 x 2-7 x-41-4 x 2 $$

Calculate \((f \cdot g)(x)\) and determine the restrictions to the domain. $$ f(x)=3 x+2 x+2 \text { and } g(x)=x 2-4(3 x+2) 2 $$

Solve for the indicated variable. Solve for \(\pi: r=C 2 \pi\).

Solve. Solve for \(y: x=2 y-13 y\).

Calculate \((f \cdot g)(x)\) and determine the restrictions to the domain. $$ f(x)=(1-3 x) 2 x-6 \text { and } g(x)=(x-6) 29 \times 2-1 $$

If \(y\) varies directly as the square of \(x\), then how does \(y\) change if \(x\) is doubled?

Use algebra to solve the following applications. A positive integer is twice another. The sum of the reciprocals of the two positive integers is \(1 / 4\). Find the two integers.

If \(y\) varies inversely as square of \(t,\) then how does \(y\) change if \(t\) is doubled?

Use algebra to solve the following applications. If the reciprocal of the smaller of two consecutive integers is subtracted from three times the reciprocal of the larger, the result is 3/10. Find the integers.

Solve for the indicated variable. Solve for \(w: P=2(l+w)\).

Simplify. $$ 1 y+1+1 y+2 y 2-1 $$

If \(y\) varies directly as the square of \(x\) and inversely as the square of \(t,\) then how does \(y\) change if both \(x\) and \(t\) are doubled?

State the restrictions and then simplify. $$ 64-x 3 x 2-8 x+16 $$

Use algebra to solve the following applications. Mary can jog, on average, 2 miles per hour faster than her husband, James. James can jog 6.6 miles in the same amount of time it takes Mary to jog 9 miles. How fast, on average, can Mary jog?

Calculate \((f / g)(x)\) and state the restrictions. $$ f(x)=1 x \text { and } g(x)=x-2 x-1 $$

Solve for the indicated variable. Solve for \(t: A=P(1+r t)\).

Use algebra to solve the following applications. Billy traveled 140 miles to visit his grandmother on the bus and then drove the 140 miles back in a rental car. The bus averages 14 miles per hour slower than the car. If the total time spent traveling was 4.5 hours, then what was the average speed of the bus?

Solve for the indicated variable. Solve for \(m: s=1 n+m .\)

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

Simplify. (Assume all denominators are nonzero.) $$ -15 x 3 y 25 x y 2(x+y) $$

Use algebra to solve the following applications. Jerry takes twice as long as Manny to assemble a skateboard. If they work together, they can assemble a skateboard in 6 minutes. How long would it take Manny to assemble the skateboard without Jerry's help?

Solve for the indicated variable. Solve for \(S: h=S 2 \pi r-r\)

Simplify. $$ 6-1+4-2 $$

Use algebra to solve the following applications. Working alone, Joe completes the yard work in 30 minutes. It takes Mike 45 minutes to complete work on the same yard. How long would it take them working together?

Simplify. $$ x-1+y-1 $$

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

Construct a mathematical model given the following. $$ y \text { varies directly with } x \text { , and } y=12 \text { when } x=4 $$