Chapter 7

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{3 x}{(x+1)^{2}}-\left[\frac{5 x+1}{(x+1)^{2}}-\frac{3 x+2}{(x+1)^{2}}\right]$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{9-15 x}{5 x^{2}-3 x}$$

Use the \([\text { GRAPH }]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the simplification is correct. If the answer is wrong, correct it and then verify your corrected simplification using the graphing utility. \(\frac{\frac{1}{x}+1}{\frac{1}{x}}=2\)

Add or subtract as indicated. Simplify the result, if possible. $$7-\frac{4 y}{y+5}$$

Perform the indicated operation or operations. $$\frac{5 x^{2}-x}{3 x+2} \div\left(\frac{6 x^{2}+x-2}{10 x^{2}+3 x-1} \cdot \frac{2 x^{2}-x-1}{2 x^{2}-x}\right)$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{b}{a c+a d-b c-b d}-\frac{a}{a c+a d-b c-b d}$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}-1}{1-x}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{9 x+3}{x^{2}-x-6}+\frac{x}{3-x}$$

Perform the indicated operation or operations. $$\frac{x^{2}+x z+x y+y z}{x-y} \div \frac{x+z}{x+y}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve the equation \(\frac{6}{x+3}=\frac{4}{x-3}\) by multiplying both sides by the LCD.

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{y}{a x+b x-a y-b y}-\frac{x}{a x+b x-a y-b y}$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}-4}{2-x}$$

Factor completely: \(2 x^{3}-20 x^{2}+50 x .\) (Section 6.5 Example 2 )

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}+9 x}{x^{2}-2 x-3}+\frac{5}{3-x}$$

Perform the indicated operation or operations $$\frac{x^{2}-x z+x y-y z}{x-y}+\frac{x-z}{y-x}$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{(y-3)(y+2)}{(y+1)(y-4)}-\frac{(y+2)(y+3)}{(y+1)(4-y)}-\frac{(y+5)(y-1)}{(y+1)(4-y)}$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{y^{2}-y-12}{4-y}$$

Solve: \(2-3(x-2)=5(x+5)-1 .\) (Section 2.3 Example 3)

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x+3}{x^{2}+x-2}-\frac{2}{x^{2}-1}$$

Perform the indicated operation or operations. $$\frac{3 x y+a y+3 x b+a b}{9 x^{2}-a^{2}} \div \frac{y^{3}+b^{3}}{6 x-2 a}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I must have made an error if a rational equation produces no solution.

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{(y+1)(2 y-1)}{(y-2)(y-3)}+\frac{(y+2)(y-1)}{(y-2)(y-3)}-\frac{(y+5)(2 y+1)}{(3-y)(2-y)}$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{y^{2}-7 y+12}{3-y}$$

Multiply: \(\quad(x+y)\left(x^{2}-x y+y^{2}\right) .\) (Section 5.2, Example 7)

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x^{2}-10 x+25}-\frac{x-4}{2 x-10}$$

Perform the indicated operation or operations. $$\frac{5 x y-a y-5 x b+a b}{25 x^{2}-a^{2}} \div \frac{y^{3}-b^{3}}{15 x+3 a}$$

Anthropologists and forensic scientists classify skulls using $$ \frac{L+60 W}{L}-\frac{L-40 W}{L} $$ where \(L\) is the skull's length and \(W\) is its width. a. Express the classification as a single rational expression. b. If the value of the rational expression in part (a) is less than \(75,\) a skull is classified as long. A medium skull has a value between 75 and \(80,\) and a round skull has a value over \(80 .\) Use your rational expression from part (a) to classify a skull that is 5 inches wide and 6 inches long.

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2} y-x^{2}}{x^{3}-x^{3} y}$$

Will help you prepare for the material covered in the next section. Solve: \(\frac{x}{3}+\frac{x}{2}=\frac{5}{6}\).

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y+3}{5 y^{2}}-\frac{y-5}{15 y}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(a\) is any real number, the equation \(\frac{a}{x}+1=\frac{a}{x}\) has no solution.

The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is modeled by $$ \frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1} $$ a. Express the temperature as a single rational expression. b. Use your rational expression from part (a) to find the temperature of the dessert, to the nearest hundredth of a degree, after 1 hour and after 2 hours.

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x y-2 x}{3 y-6}$$

Will help you prepare for the material covered in the next section. Solve: \(\frac{2 x}{3}=\frac{14}{3}-\frac{x}{2}\).

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-7}{3 y^{2}}-\frac{y-2}{12 y}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. All real numbers satisfy the equation \(\frac{3}{x}-\frac{1}{x}=\frac{2}{x}\)

Will help you prepare for the material covered in the next section. Solve: \(2 x^{2}+2=5 x\).

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}+2 x y-3 y^{2}}{2 x^{2}+5 x y-3 y^{2}}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x+3}{3 x+6}+\frac{x}{4-x^{2}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. To solve \(\frac{5}{3 x}+\frac{3}{x}=1,\) we must first add the rational expressions on the left side.

The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is modeled by $$ \frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1} $$ a. Express the temperature as a single rational expression. b. Use your rational expression from part (a) to find the temperature of the dessert, to the nearest hundredth of a degree, after 1 hour and after 2 hours.

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}+3 x y-10 y^{2}}{3 x^{2}-7 x y+2 y^{2}}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x+7}{4 x+12}+\frac{x}{9-x^{2}}$$

$$\text { Solve for } f: \frac{1}{p}+\frac{1}{q}=\frac{1}{f}$$

Explain how to add rational expressions when denominators are the same. Give an example with your explanation.

Simplify each rational expression. $$\frac{x^{2}-9 x+18}{x^{3}-27}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y^{2}-1}+\frac{2 y}{y-y^{2}}$$

Explain how to multiply rational expressions.

$$\text { Solve for } f_{2}: f=\frac{f_{1} f_{2}}{f_{1}+f_{2}}$$

Explain how to subtract rational expressions when denominators are the same. Give an example with your explanation.