8.13: Proficiency Exam
Proficiency Exam
Find the domain of \(\dfrac{5a + 1}{a^2 - 5a - 24}\)
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\(a≠−3,8\)
For the following problems, fill in the missing term.
\(-\dfrac{3}{x+4} = \dfrac{?}{x + 4}\)
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\(−3\)
\(\dfrac{2x + 5}{-x + 1} = \dfrac{?}{x - 1}\)
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\(−2x−5\)
For the following problems, reduce to lowest terms.
\(\dfrac{30x^6y^3(x-3)^2(x+5)^2}{6xy^3(x+5)}\)
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\(5x^5(x-3)^2(x+5)\)
\(\dfrac{x^2 + 10x + 24}{x^2 + x - 30}\)
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\(\dfrac{x + 4}{x - 5}\)
\(\dfrac{8x^2 + 2x - 3}{4x^2 + 12x - 7}\)
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\(\dfrac{4x + 3}{2x + 7}\)
Replace \(N\) with the proper quantity.
\(\dfrac{x+2}{x-1} = \dfrac{N}{x^2 - 4x + 3}\)
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\((x−3)(x+2)\)
Assume that \(a^2 + a - 6, a^2 - a - 12\), and \(a^2 - 2a - 8\) are denominators of rational expressions. Find the LCD.
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\((a+2)(a−2)(a+3)(a−4)\)
For the following problems, perform the operations.
\(\dfrac{3a + 4}{a + 6} - \dfrac{2a - 1}{a + 6}\)
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\(\dfrac{a+5}{a+6}\)
\(\dfrac{18x^3y}{5a^2} \cdot \dfrac{15a^3b}{6x^2y}\)
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\(9abx\)
\(\dfrac{y^2-y-12}{y^2 + 3y + 2} \cdot \dfrac{y^2 + 10y + 16}{y^2 - 7y + 12}\)
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\(\dfrac{(y+3)(y+8)}{(y+1)(y-3)}\)
\(\dfrac{y-2}{y^2 - 11y + 24} + \dfrac{y + 4}{y^2 + 3y - 18}\)
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\(\dfrac{2(y^2 - 22)}{(y-8)(y-3)(y+6)}\)
\(\dfrac{9}{2x + 7} + \dfrac{4}{6x - 1}\)
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\(\dfrac{62x + 19}{(2x + 7)(6x - 1)}\)
\(\dfrac{16x^5(x^2 - 1)}{9x - 9} \div \dfrac{2x^2 - 2x}{3}\)
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\(\dfrac{8x^4(x + 1)}{3(x-1)}\)
\((m + 3) \div \dfrac{2m + 6}{5m + 1}\)
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\(\dfrac{5m + 1}{2}\)
\(\dfrac{3y + 10}{8y^2 + 10y - 3} - \dfrac{5y - 1}{4y^2 + 23y - 6}\)
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\(\dfrac{-7y^2 + 15y + 63}{(4y-1)(2y + 3)(y + 6)}\)
Solve \(\dfrac{1}{x+3} + \dfrac{3}{x-3} = \dfrac{x}{x^2 - 9}\)
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\(x = -2\)
Solve \(\dfrac{12}{m-4} + 5 = \dfrac{3m}{m-4}\).
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No solution; \(m=4\) is excluded.
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?
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\(7\)
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?
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12 hours
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.
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8 ft by 14 ft
Simplify the complex fraction \(\dfrac{4 - \frac{3}{x}}{4 + \frac{3}{x}}\)
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\(\dfrac{4x - 3}{4x + 3}\)
Simplify the complex fraction \(\dfrac{1-\frac{5}{x}-\frac{6}{x^{2}}}{1+\frac{6}{x}+\frac{5}{x^{2}}}\)
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\(\dfrac{x-6}{x + 5}\)
Perform the division: \(\dfrac{x^3 + 10x^2 + 21x - 18}{x + 6}\)
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\(x^2 + 4x - 3\)
Perform the division: \(\dfrac{2x^3 + 5x - 1}{x - 2}\)
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\(2x^2 + 4x + 13 + \dfrac{25}{x - 2}\)