Question

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

\\_\\_\\_\\_\\_\

1. \\(\frac{x + 2}{x - 1}-\frac{x^{2}+6x + 14}{x^{2}+5x - 6}\\)\

\\_\\_\\_\\_\\_\

1. \\(\frac{30x^{3}}{6x}\\)\

a. \\(5x^{2}\\) b. \\(5x\\)\
c. 24 d. \\(24x^{2}\\)\

1. \\(\frac{12x - 30}{12}\\)\

a. \\(2x - 5\\) b. \\(\frac{2x}{2}\\)\
c. \\(\frac{2x - 5}{12}\\) d. \\(\frac{2x - 5}{2}\\)\

1. \\(\frac{28x^{3}y^{2}}{4x^{10}y}\\)\

a. \\(\frac{7x^{7}}{y}\\) b. \\(\frac{7y}{x^{7}}\\)\
c. \\(7x^{7}y^{2}\\) d. \\(7xy\\)\

1. \\(\frac{2x^{2}+2x}{14x^{2}+6x}\\)\

a. \\(\frac{x^{2}+1}{7x^{2}+3}\\) b. \\(\frac{x + 2}{7x + 3}\\)\
c. \\(\frac{x + 1}{7x + 3}\\) d. \\(\frac{x + 1}{x + 3}\\)

Explanation:

Step1: Factor denominator (Q9)

Factor $x^2-5x+6=(x-2)(x-3)$

Step2: Get common denominator (Q9)

$\frac{x}{x-2} = \frac{x(x-3)}{(x-2)(x-3)}$

Step3: Add fractions (Q9)

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

Step4: Simplify numerator (Q9)

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

Step5: Cancel common terms (Q9)

$\frac{x+2}{x-3}$
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Step1: Factor denominator (Q10)

Factor $x^2+5x-6=(x-1)(x+6)$

Step2: Get common denominator (Q10)

$\frac{x+2}{x-1} = \frac{(x+2)(x+6)}{(x-1)(x+6)}$

Step3: Subtract fractions (Q10)

$\frac{(x+2)(x+6)-(x^2+6x+14)}{(x-1)(x+6)}$

Step4: Expand numerator (Q10)

$\frac{x^2+8x+12-x^2-6x-14}{(x-1)(x+6)}$

Step5: Simplify numerator (Q10)

$\frac{2x-2}{(x-1)(x+6)} = \frac{2(x-1)}{(x-1)(x+6)}$

Step6: Cancel common terms (Q10)

$\frac{2}{x+6}$
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Step1: Cancel coefficients (Q11)

$\frac{30}{6}=5$

Step2: Simplify variable terms (Q11)

$\frac{x^3}{x}=x^2$

Step3: Combine results (Q11)

$5x^2$
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Step1: Factor numerator (Q12)

$12x-30=6(2x-5)$

Step2: Cancel common factor (Q12)

$\frac{6(2x-5)}{12}=\frac{2x-5}{2}$
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Step1: Cancel coefficients (Q13)

$\frac{28}{4}=7$

Step2: Simplify $x$-terms (Q13)

$\frac{x^3}{x^{10}}=x^{-7}=\frac{1}{x^7}$

Step3: Simplify $y$-terms (Q13)

$\frac{y^2}{y}=y$

Step4: Combine results (Q13)

$\frac{7y}{x^7}$
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Step1: Factor numerator/denominator (Q14)

Numerator: $2x^2+2x=2x(x+1)$; Denominator: $14x^2+6x=2x(7x+3)$

Step2: Cancel common terms (Q14)

$\frac{2x(x+1)}{2x(7x+3)}=\frac{x+1}{7x+3}$

Answer:

1. $\boldsymbol{\frac{x+2}{x-3}}$
2. $\boldsymbol{\frac{2}{x+6}}$
3. A. $5x^2$
4. D. $\frac{2x-5}{2}$
5. B. $\frac{7y}{x^7}$
6. C. $\frac{x+1}{7x+3}$