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\(f(x)=\frac{(x-4)(x+1)}{(x-2)(x-4)}\)

The domain of \(f\) is \((-\infty,2) \cup (2,4) \cup (4,\infty)\text{.}\)

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State the domain of each function using interval notation.

\(f(x)=\frac{(x-4)(x+1)}{(x-2)(x-4)}\)

Solution

The domain of \(f\) is \((-\infty,2) \cup (2,4) \cup (4,\infty)\text{.}\)

\(g(x)=\frac{x^2-5x-14}{x^2+9x+20}\)

Solution

The domain of \(g\) is \((-\infty,-5) \cup (-5,-4) \cup (-4,\infty)\text{.}\)

\(h(x)=\frac{x^2+9}{3x^3-75x}\)

Solution

The domain of \(h\) is \((-\infty,-5) \cup (-5,0) \cup (0,5) \cup (5,\infty)\text{.}\)

Simplify each expression. Make sure that you state any necessary domain restrictions.

\(\frac{x^2-5x-14}{x^2-49}\)

Solution

\(\frac{x^2-5x-14}{x^2-49}=\frac{x+2}{x+7}\text{,}\) \(x \neq 7\)

\(\frac{x^3+9x}{x^3-9x}\)

Solution

\(\frac{x^3+9x}{x^3-9x}=\frac{x^2+9}{(x-3)(x+3)}\text{,}\) \(x \neq 0\)

\(\frac{x^2-15x+54}{9-x}\)

Solution

\(\frac{x^2-15x+54}{9-x}=6-x\text{,}\) \(x \neq 9\)

Multiply or divide as indicated. Completely simplify each result and state any necessary domain restrictions.

\(\frac{x+6}{x-7} \cdot \frac{2x-14}{x+12}\)

Solution

\(\frac{x+6}{x-7} \cdot \frac{2x-14}{x+12}=\frac{2(x+6)}{x+12}\text{,}\) \(x \neq 7\)

\(\frac{x}{x-8} \div \frac{x^3-64x}{x+12}\)

Solution

\(\frac{x}{x-8} \div \frac{x^3-64x}{x+12}=\frac{x+12}{(x+8)(x-8)^2}\text{,}\) \(x \neq 0\)

\((x+3) \div \frac{x^2+9}{x+3}\)

Solution

\((x+3) \div \frac{x^2+9}{x+3}=\frac{(x+3)^2}{x^2+9}\)

\(\frac{x^2-8x+16}{x} \div (x^2-3x-4)\)

Solution

\(\frac{x^2-8x+16}{x} \div (x^2-3x-4)=\frac{x-4}{x(x+1)}\text{,}\) \(x \neq 4\)

Add or subtract as indicated. Completely simplify each result and state any necessary domain restrictions.

\(\frac{x^2}{x^2+2x+1}-\frac{3x+4}{x^2+2x+1}\)

Solution

\(\frac{x^2}{x^2+2x+1}-\frac{3x+4}{x^2+2x+1}=\frac{x-4}{x+1}\)

\(\frac{2x+7}{x+4}-\frac{x-5}{x-4}\)

Solution

\(\frac{2x+7}{x+4}-\frac{x-5}{x-4}=\frac{x^2-8}{(x-4)(x+4)}\)

\(\frac{x}{x-14}+\frac{14}{14-x}\)

Solution

\(\frac{x}{x-14}+\frac{14}{14-x}=1\text{,}\) \(x \neq 14\)

\(\frac{x+1}{x^2-5x+6}+\frac{x+1}{x^2-4x+4}\)

Solution

\(\frac{x+1}{x^2-5x+6}+\frac{x+1}{x^2-4x+4}=\frac{(x+1)(2x-5)}{(x-3)(x-2)^2}\)

Completely simplify each complex fraction. You do not need to address any domain restrictions.

\(\frac{\frac{21}{x+7}}{\frac{3}{x+7}}\)

Solution

\(\frac{\frac{21}{x+7}}{\frac{3}{x+7}}=7\)

\(\frac{\frac{8}{x^2-4}}{\frac{16x+24}{x-2}}\)

Solution

\(\frac{\frac{8}{x^2-4}}{\frac{16x+24}{x-2}}=\frac{1}{(2x+3)(x+2)}\)

\(\frac{\frac{1}{x+2}+\frac{1}{x-1}}{\frac{1}{x-1}}\)

Solution

\(\frac{\frac{1}{x+2}+\frac{1}{x-1}}{\frac{1}{x-1}}=\frac{2x+1}{x+2}\)

\(\frac{3}{\frac{x}{x+4}-\frac{2}{x}}\)

Solution

\(\frac{3}{\frac{x}{x+4}-\frac{2}{x}}=\frac{3x(x+4)}{(x-4)(x+2)}\)

\(\frac{\frac{1}{x+8}-\frac{1}{x}}{x}\)

Solution

\(\frac{\frac{1}{x+8}-\frac{1}{x}}{x}=-\frac{8}{x^2(x+8)}\)

Determine the solution set to each equation.

\(\frac{3}{2x+1}=\frac{5}{4x-3}\)

Solution

The solution set is \(\{7\}\text{.}\)

\(\frac{x}{x+4}+\frac{1}{x-2}=1\)

Solution

The solution set is \(\{4\}\text{.}\)

\(\frac{6}{x^2-9x+14}-\frac{4}{x-7}=\frac{3}{x-2}\)

Solution

The solution set is \(\{5\}\text{.}\)

\(x-\frac{7x}{x-2}=1-\frac{14}{x-2}\)

Solution

The solution set is \(\{8\}\text{.}\)

Determine the solution for each application problem.

Two trains are cruising through the plains of Canada. One train is westbound and the other eastbound. They both travel at constant speeds with the westbound train traveling at a rate that is 20 km/hr faster than the rate at which the eastbound train is traveling. Over a certain period of time, the faster train travels 265 km while the slower train travels only 215 km. What is the constant speed of each train?

Solution

The faster (westbound) train moves at the speed of \(106\) km/hr while the slower (eastbound) train moves at a speed of \(86\) km/hr.

Trujillo home improvement has been awarded the contract for a small bathroom remodel. Working alone, Pedro, who has been working for the company for several years, could get the job done in \(8\) hours. Working alone, it would take newly hired Julia \(12\) hours to complete the job. Assuming that they can maintain their individual paces, how long would it take Pedro and Julia to complete the job if they work together?

Solution

Working together, Pedro and Julia can complete the job in \(4.8\) hours.