8.12 Proficiency exam

( [link] ) Find the domain of $\frac{5a+1}{a^2-5a-24}.$

For the following problems, fill in the missing term.
( [link] ) $-\frac{3}{x+4}=\frac{}{x+4}$

( [link] ) $\frac{2x+5}{-x+1}=\frac{}{x-1}$

For the following problems, reduce to lowest terms.
( [link] ) $\frac{30x^6{y}^3{\left(x-3\right)}^2{\left(x+5\right)}^2}{6x{y}^3\left(x+5\right)}$

( [link] ) $\frac{x^2+10x+24}{x^2+x-30}$

( [link] ) $\frac{8x^2+2x-3}{4x^2+12x-7}$

( [link] ) Replace $N$ with the proper quantity.
$\frac{x+2}{x-1}=\frac{N}{x^2-4x+3}$

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

For the following problems, perform the operations.
( [link] ) $\frac{3a+4}{a+6}-\frac{2a-1}{a+6}$

( [link] ) $\frac{18x^{3}y}{5a^2}·\frac{15a^{3}b}{6x^{2}y}$

( [link] ) $\frac{y^2-y-12}{y^2+3y+2}·\frac{y^2+10y+16}{y^2-7y+12}$

( [link] ) $\frac{y-2}{y^2-11y+24}+\frac{y+4}{y^2+3y-18}$

( [link] ) $\frac{9}{2x+7}+\frac{4}{6x-1}$

( [link] ) $\frac{16x^5\left(x^2-1\right)}{9x-9}÷\frac{2x^2-2x}{3}$

( [link] ) $\left(m+3\right)÷\frac{2m+6}{5m+1}$

( [link] ) $\frac{3y+10}{8y^2+10y-3}-\frac{5y-1}{4y^2+23y-6}$

( [link] ) Solve $\frac{1}{x+3}+\frac{3}{x-3}=\frac{x}{x^2-9}.$

( [link] ) Solve $\frac{12}{m-4}+5=\frac{3m}{m-4}.$

( [link] ) When the same number is added to both the numerator and denominator of the fraction $\frac{5}{3},$ the result is $\frac{6}{5}.$ What is the number that is added?

( [link] ) 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?

( [link] ) 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.

( [link] ) Simplify the complex fraction $\frac{4-\frac{3}{x}}{4+\frac{3}{x}}.$

( [link] ) Simplify the complex fraction $\frac{1-\frac{5}{x}-\frac{6}{x^2}}{1+\frac{6}{x}+\frac{5}{x^2}}.$

( [link] ) Perform the division: $\frac{x^3+10x^2+21x-18}{x+6}.$

( [link] ) Perform the division: $\frac{2x^3+5x-1}{x-2}.$