8.13: Proficiency Exam

Proficiency Exam

Exercise $\PageIndex{1}$

Find the domain of $\dfrac{5a + 1}{a^2 - 5a - 24}$

Answer

$a≠−3,8$

For the following problems, fill in the missing term.

Exercise $\PageIndex{2}$

$-\dfrac{3}{x+4} = \dfrac{?}{x + 4}$

Answer

$−3$

Exercise $\PageIndex{3}$

$\dfrac{2x + 5}{-x + 1} = \dfrac{?}{x - 1}$

Answer

$−2x−5$

For the following problems, reduce to lowest terms.

Exercise $\PageIndex{4}$

$\dfrac{30x^6y^3(x-3)^2(x+5)^2}{6xy^3(x+5)}$

Answer

$5x^5(x-3)^2(x+5)$

Exercise $\PageIndex{5}$

$\dfrac{x^2 + 10x + 24}{x^2 + x - 30}$

Answer

$\dfrac{x + 4}{x - 5}$

Exercise $\PageIndex{6}$

$\dfrac{8x^2 + 2x - 3}{4x^2 + 12x - 7}$

Answer

$\dfrac{4x + 3}{2x + 7}$

Replace $N$ with the proper quantity.

Exercise $\PageIndex{7}$

$\dfrac{x+2}{x-1} = \dfrac{N}{x^2 - 4x + 3}$

Answer

$(x−3)(x+2)$

Exercise $\PageIndex{8}$

Assume that $a^2 + a - 6, a^2 - a - 12$, and $a^2 - 2a - 8$ are denominators of rational expressions. Find the LCD.

Answer

$(a+2)(a−2)(a+3)(a−4)$

For the following problems, perform the operations.

Exercise $\PageIndex{9}$

$\dfrac{3a + 4}{a + 6} - \dfrac{2a - 1}{a + 6}$

Answer

$\dfrac{a+5}{a+6}$

Exercise $\PageIndex{10}$

$\dfrac{18x^3y}{5a^2} \cdot \dfrac{15a^3b}{6x^2y}$

Answer

$9abx$

Exercise $\PageIndex{11}$

$\dfrac{y^2-y-12}{y^2 + 3y + 2} \cdot \dfrac{y^2 + 10y + 16}{y^2 - 7y + 12}$

Answer

$\dfrac{(y+3)(y+8)}{(y+1)(y-3)}$

Exercise $\PageIndex{12}$

$\dfrac{y-2}{y^2 - 11y + 24} + \dfrac{y + 4}{y^2 + 3y - 18}$

Answer

$\dfrac{2(y^2 - 22)}{(y-8)(y-3)(y+6)}$

Exercise $\PageIndex{13}$

$\dfrac{9}{2x + 7} + \dfrac{4}{6x - 1}$

Answer

$\dfrac{62x + 19}{(2x + 7)(6x - 1)}$

Exercise $\PageIndex{14}$

$\dfrac{16x^5(x^2 - 1)}{9x - 9} \div \dfrac{2x^2 - 2x}{3}$

Answer

$\dfrac{8x^4(x + 1)}{3(x-1)}$

Exercise $\PageIndex{15}$

$(m + 3) \div \dfrac{2m + 6}{5m + 1}$

Answer

$\dfrac{5m + 1}{2}$

Exercise $\PageIndex{16}$

$\dfrac{3y + 10}{8y^2 + 10y - 3} - \dfrac{5y - 1}{4y^2 + 23y - 6}$

Answer

$\dfrac{-7y^2 + 15y + 63}{(4y-1)(2y + 3)(y + 6)}$

Exercise $\PageIndex{17}$

Solve $\dfrac{1}{x+3} + \dfrac{3}{x-3} = \dfrac{x}{x^2 - 9}$

Answer

$x = -2$

Exercise $\PageIndex{18}$

Solve $\dfrac{12}{m-4} + 5 = \dfrac{3m}{m-4}$.

Answer

No solution; $m=4$ is excluded.

Exercise $\PageIndex{19}$

When the same number is added to both the numerator and denominator of the fraction $\dfrac{5}{3}$, the result is $\dfrac{6}{5}$. What is the number that is added?

Answer

$7$

Exercise $\PageIndex{20}$

Person A, working alone, can complete a job in 20 hours. Person B, working alone, can complete the same job in 30 hours. How long will it take both people, working together, to complete the job?

Answer

12 hours

Exercise $\PageIndex{21}$

The width of a rectangle is 1 foot longer than one-half the length. Find the dimensions (lengh and width) of the rectangle if the perimeter is 44 feet.

Answer

8 ft by 14 ft

Exercise $\PageIndex{22}$

Simplify the complex fraction $\dfrac{4 - \frac{3}{x}}{4 + \frac{3}{x}}$

Answer

$\dfrac{4x - 3}{4x + 3}$

Exercise $\PageIndex{23}$

Simplify the complex fraction $\dfrac{1-\frac{5}{x}-\frac{6}{x^{2}}}{1+\frac{6}{x}+\frac{5}{x^{2}}}$

Answer

$\dfrac{x-6}{x + 5}$

Exercise $\PageIndex{24}$

Perform the division: $\dfrac{x^3 + 10x^2 + 21x - 18}{x + 6}$

Answer

$x^2 + 4x - 3$

Exercise $\PageIndex{25}$

Perform the division: $\dfrac{2x^3 + 5x - 1}{x - 2}$

Answer

$2x^2 + 4x + 13 + \dfrac{25}{x - 2}$