Add & Subtract Rational Expressions with Unlike Denominators 1

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Question 1 of 4

Perform the operation

\frac{3 x}{x - 8} + \frac{5 x}{8 - x}

$(3x)/(x-8)+(5x)/(8-x)$

1. $\frac{x - 8}{2 x}$ $(x-8)/(2x)$
2. $\frac{2 x}{x - 8}$ $(2x)/(x-8)$
3. $\frac{- 2 x}{x - 8}$ $(-2x)/(x-8)$
4. $x - 8$ $x-8$

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Adding and Subtracting Rational Expressions

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

$a/c +- b/c = (a+-b)/c$

Rewrite the expressions such that they have the same denominators.

$x - 8$ $x-8$	$=$ $=$	$- (8 - x)$ $-(8-x)$

$\frac{3x}{x-8}+ \frac{5x}{8-x}$	$=$	$\frac{3x}{x-8}+ \frac{5x \times \color {red}{-1}}{(8-x) \times \color{red}{-1}}$	Change $8 - x$ $8-x$ to $x - 8$ $x-8$

	$=$ $=$	$\frac{3 x}{x - 8} - \frac{5 x}{x - 8}$ $(3x)/(x-8) – (5x)/(x-8)$	Simplify

	$=$ $=$	$\frac{3 x - 5 x}{x - 8}$ $(3x-5x)/(x-8)$	Combine the numerators

	$=$ $=$	$\frac{- 2 x}{x - 8}$ $(-2x)/(x-8)$

\frac{- 2 x}{x - 8}

$(-2x)/(x-8)$

Question 2 of 4

Perform the operation

\frac{13}{n^{2} - 16} + \frac{- 5}{4 - n}

$(13)/(n^2-16)+(-5)/(4-n)$

1. $\frac{5 n + 33}{n - 4}$ $(5n+33)/(n-4)$
2. $\frac{5 n + 33}{n^{2} - 16}$ $(5n+33)/(n^2-16)$
3. $n + 4$ $n+4$
4. $5 n + 17$ $5n+17$

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Adding and Subtracting Rational Expressions

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

$a/c +- b/c = (a+-b)/c$

Expand the denominator of the first term.

$\frac{13}{n^{2} - 16} + \frac{- 5}{4 - n}$ $(13)/(n^2-16)+(-5)/(4-n)$	$=$ $=$	$\frac{13}{(n - 4) (n + 4)} + \frac{- 5}{4 - n}$ $(13)/((n-4)(n+4))+(-5)/(4-n)$	$n^{2} - 16 = (n - 4) (n + 4)$ $n^2-16 = (n-4)(n+4)$

	$=$	$(13)/((n-4)(n+4))+(5)/(n-4)$	$4-n = – (n-4)$

Multiply the second term by

$(n+4)/(n+4)$

$\frac {13}{n^2-16}+ \frac{-5}{4-n}$	$=$	$\frac{13}{(n-4)(n+4)}+ \frac{5 \times \color{red}{(n+4)}}{(n-4) \times \color {red}{(n+4)}}$

	$=$	$(13)/((n-4)(n+4))+(5(n+4))/((n-4))(n+4)$	Simplify

	$=$	$(13 +5(n+4))/((n-4)(n+4))$	Combine the numerators

	$=$	$(13 +5n + 20)/((n-4)(n+4))$	Distribute $5$ inside parenthesis

	$=$	$(5n+33)/((n-4)(n+4))$	Simplify

	$=$	$(5n+33)/(n^2-16)$

$(5n+33)/(n^2-16)$

Question 3 of 4

Perform the operation

$(5x)/(8y)+(3x)/(24y^2)$

1. $(5xy-x)/(8y^2)$
2. $(5xy+x)/(8y^2)$
3. $(15xy-3x)/(8y^2)$
4. $5xy-x$

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Adding and Subtracting Rational Expressions

$a/c +- b/c = (a+-b)/c$

Rewrite the expressions such that they have the same denominators.

$(5x)/(8y)-(3x)/(24y^2)$	$=$	$(5x)/(8y)+(3x)/((8y)(3y))$	$24y^2 = (8y)(3y)$

	$=$	$\frac{5x \times \color{red}{3y}}{8y \times \color {red}{3y}}- \frac {3x}{(8y)(3y)}$	Multiply the first term by $(3y)/(3y)$

	$=$	$(15xy)/((8y)(3y))-(3x)/((8y)(3y))$	Simplify

	$=$	$(15xy-3x)/((8y)(3y))$

	$=$	$(3(5xy-x))/((8y)(3y))$	Factor out $3$ from numerator

	$=$	$(5xy-x)/((8y)y)$	Cancel out $3$

	$=$	$(5xy-x)/(8y^2)$	Simplify

$(5xy-x)/(8y^2)$

Question 4 of 4

Perform the operation

$1/(a-1)-1/a$

1. $a-1$
2. $1$
3. $a(a-1)$
4. $1/(a(a-1))$

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Adding and Subtracting Rational Expressions

$a/c +- b/c = (a+-b)/c$

First, identify the LCD.

Since we have different denominators, the LCD would be

$(a)(a-1)$ .

Convert into same denominators using the LCD.

$\frac {1}{a-1}- \frac {1}{a}$	$=$	$\frac {1 \times \color{red}{a}}{(a-1) \times \color {red}{a}}- \frac {1 \times \color{red}{(a-1)}}{a \times \color{red}{(a-1)}}$	LCD is $a(a-1)$

	$=$	$a/(a(a-1))-(a-1)/(a(a-1))$	Simplify

	$=$	$(a-(a-1))/(a(a-1))$	Combine the numerators

	$=$	$(a-a+1)/(a(a-1))$	Distribute the negative sign

	$=$	$1/(a(a-1))$

$1/(a(a-1))$

Add & Subtract Rational Expressions with Unlike Denominators 1

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Adding and Subtracting Rational Expressions

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