Chapter 7

Solve for the indicated variable. Solve for \(x: y=2 x+15 x\).

Simplify. $$ x-2-y-1 $$

Construct a mathematical model given the following. $$ y \text { varies inversely as } x, \text { and } y=2 \text { when } x=5 $$

Simplify. $$ (2 x-1)-1-x-2 $$

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

Construct a mathematical model given the following. $$ y \text { is jointly proportional to } x \text { and } z \text { , where } y=36 \text { when } x=3 \text { and } z=4 \text { . } $$

Simplify. $$ (x-4)-1-(x+1)-1 $$

Simplify. (Assume all denominators are nonzero.) $$ a 2-a b-6 b 2 a 2-6 a b+9 b 2 $$

Construct a mathematical model given the following. \(y\) is directly proportional to the square of \(x\) and inversely proportional to \(z,\) where \(y=20\) when \(x=2\) and \(z=5\).

Explain why multiplying both sides of an equation by the LCD sometimes produces extraneous solutions.

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

Simplify. (Assume all denominators are nonzero.) $$ 2 a 2-11 a+12-32+2 a 2 $$

The distance an object in free fall drops varies directly with the square of the time that it has been falling. It is observed that an object falls 16 feet in 1 second. Find an equation that models the distance an object will fall and use it to determine how far it will fall in 2 seconds.

Explain why \(x=7\) is a restriction to \(1 x \div x-7 x-2\).

Explain the connection between the technique of cross multiplication and multiplying both sides of a rational equation by the LCD.

Simplify. $$ 2(y-1)-2-(y-1)-1 $$

The weight of an object varies inversely as the square of its distance from the center of earth. If an object weighs 180 pounds on the surface of earth (approximately 4,000 miles from the center), then how much will it weigh at 2,000 miles above earth's surface?

Explain how we can tell the difference between a rational expression and a rational equation. How do we treat them differently?

Calculate \((f+g)(x)\) and \((f-g)(x)\) and state the restrictions to the domain. \(f(x)=13 x\) and \(g(x)=1 x-2\)

Calculate \((f+g)(x)\) and \((f-g)(x)\) and state the restrictions to the domain. \(f(x)=1 x-1\) and \(g(x)=1 x+5\)

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

Calculate \((f+g)(x)\) and \((f-g)(x)\) and state the restrictions to the domain. \(f(x)=x x-5\) and \(g(x)=12 x-3\)

Calculate \((f+g)(x)\) and \((f-g)(x)\) and state the restrictions to the domain. \(f(x)=5 x+2\) and \(g(x)=3 x+4\)

Calculate \((f+f)(x)\) and state the restrictions to the domain. $$ f(x)=1 x $$

Calculate \((f+f)(x)\) and state the restrictions to the domain. $$ f(x)=12 x $$

Calculate \((f+f)(x)\) and state the restrictions to the domain. $$ f(x)=1 x+2 $$

Explain to a classmate why this is incorrect: \(1 \times 2+2 \times 2=32 \times 2 .\)

Explain to a classmate how to find the common denominator when adding algebraic expressions. Give an example.

State the restrictions to the domain and then simplify. $$ f(x)=x 2+6 x+92 x 2+5 x-3 $$

State the restrictions to the domain and then simplify. $$ g(x)=x 3-273-x $$

State the restrictions to the domain and then simplify. $$ g(x)=3 x-1510-2 x $$

The cost in dollars of producing coffee mugs with a company logo is given by \(C(x)=x+40,\) where \(x\) represents the number of mugs produced. Calculate the average cost of producing 100 mugs and the average cost of producing 500 mugs.

The cost in dollars of renting a moving truck for the day is given by \(C(x)=0.45 x+90,\) where \(x\) represents the number of miles driven. Calculate the average cost per mile if the truck is driven 250 miles in one day.

The cost in dollars of producing sweat shirts with a custom design on the back is given by \(C(x)=1200+(12-0.05 x) x\), where \(x\) represents the number of sweat shirts produced. Calculate the average cost of producing 150 custom sweat shirts.

The cost in dollars of producing a custom injected molded part is given by \(C(x)=500+(3-0.001 x) x,\) where \(x\) represents the number of parts produced. Calculate the average cost of producing 1,000 custom parts.