Add and Subtract Algebraic Fractions

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Question 1 of 7

1. Question
Perform the operation

$\frac{4 x}{14} + \frac{6 x}{14}$ $(4x)/14 + (6x)/14$
- 1. $\frac{5 x}{7}$ $(5x)/7$
- 2. $- \frac{5 x}{7}$ $-(5x)/7$
- 3. $5 x$ $5x$
- 4. $35$ $35$
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Adding and Subtracting Rational Expressions

$\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}$ $a/c +- b/c = (a+-b)/c$

Since the denominators are the same, we need to perform the operation on the numerators alone.

$\frac{4 x}{14} + \frac{6 x}{14}$ $(4x)/14 +(6x)/14$ $=$ $=$ $\frac{10 x}{14}$ $(10x)/14$

$=$ $=$ $\frac{5 x}{7}$ $(5x)/7$ Express in lowest terms

$\frac{5 x}{7}$ $(5x)/7$

Question 2 of 7

Perform the operation

\frac{4 m}{m - 1} - \frac{m + 3}{m - 1}

$(4m)/(m-1) – (m+3)/(m-1)$

1. $3 m - 3$ $3m-3$
2. $3$ $3$
3. $1$ $1$
4. $m - 2$ $m-2$

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Adding and Subtracting Rational Expressions

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

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

Since the denominators are the same, we need to perform the operation on the numerators alone.

$\frac{4 m}{m - 1} - \frac{m + 3}{m - 1}$ $(4m)/(m-1) – (m+3)/(m-1)$	$=$ $=$	$\frac{4 m - (m + 3)}{m - 1}$ $(4m-(m+3))/(m-1)$

	$=$ $=$	$\frac{4 m - m - 3}{m - 1}$ $(4m-m-3)/(m-1)$	Distribute the negative sign

	$=$ $=$	$\frac{3 m - 3}{m - 1}$ $(3m-3)/(m-1)$	Combine like terms

	$=$ $=$	$\frac{3 (m - 1)}{m - 1}$ $(3(m-1))/(m-1)$	Factor out $3$ $3$ from the numerator

	$=$ $=$	$3$ $3$	Cancel out $m - 1$ $m-1$

3

$3$

Question 3 of 7

Perform the operation

\frac{2 m + 15}{m - 3} - \frac{m - 1}{m - 3}

$(2m+15)/(m-3) – (m-1)/(m-3)$

1. $\frac{m + 16}{m - 3}$ $(m+16)/(m-3)$
2. $m + 16$ $m+16$
3. $\frac{m - 16}{m - 3}$ $(m-16)/(m-3)$
4. $\frac{3 m + 16}{m - 3}$ $(3m+16)/(m-3)$

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Adding and Subtracting Rational Expressions

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

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

Since the denominators are the same, we need to perform the operation on the numerators alone.

$\frac{2 m + 15}{m - 3} - \frac{m - 1}{m - 3}$ $(2m+15)/(m-3) – (m-1)/(m-3)$	$=$ $=$	$\frac{2 m + 15 - (m - 1)}{m - 3}$ $(2m+15 – (m-1))/(m-3)$

	$=$ $=$	$\frac{2 m + 15 - m + 1}{m - 3}$ $(2m+15-m+1)/(m-3)$	Distribute negative sign

	$=$ $=$	$\frac{m + 16}{m - 3}$ $(m+16)/(m-3)$	Simplify

\frac{m + 16}{m - 3}

$(m+16)/(m-3)$

Question 4 of 7

Perform the operation

\frac{a - 9}{2 a + 7} - \frac{- 2 a + 4}{2 a + 7}

$(a-9)/(2a+7) – (-2a+4)/(2a+7)$

1. $2 a + 7$ $2a+7$
2. $3 a - 13$ $3a-13$
3. $\frac{3 a + 13}{2 a + 7}$ $(3a+13)/(2a+7)$
4. $\frac{3 a - 13}{2 a + 7}$ $(3a-13)/(2a+7)$

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Adding and Subtracting Rational Expressions

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

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

Since the denominators are the same, we need to perform the operation on the numerators alone.

$\frac{a - 9}{2 a + 7} - \frac{- 2 a + 4}{2 a + 7}$ $(a-9)/(2a+7) – (-2a+4)/(2a+7)$	$=$ $=$	$\frac{a - 9 - (- 2 a + 4)}{2 a + 7}$ $(a-9 – (-2a+4))/(2a+7)$

	$=$ $=$	$\frac{a - 9 + 2 a - 4}{2 a + 7}$ $(a-9+2a-4)/(2a+7)$	Distribute negative sign

	$=$ $=$	$\frac{3 a - 13}{2 a + 7}$ $(3a-13)/(2a+7)$	Simplify

\frac{3 a - 13}{2 a + 7}

$(3a-13)/(2a+7)$

Question 5 of 7

Perform the operation

4 + \frac{7}{x + 5}

$4+7/(x+5)$

1. $4 x + 20$ $4x+20$
2. $4$ $4$
3. $\frac{4 x + 27}{x - 5}$ $(4x+27)/(x-5)$
4. $\frac{4 x + 27}{x + 5}$ $(4x+27)/(x+5)$

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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.

$4 + \frac{7}{x + 5}$ $4+7/(x+5)$	$=$ $=$	$\frac{4}{1} + \frac{7}{x + 5}$ $4/1+7/(x+5)$	Convert $4$ $4$ into a fraction

	$=$	$\frac{4 \times \color{red}{(x+5)}}{1 \times \color {red}{(x+5)}} + \frac{7}{x+5}$	Multiply $\frac{4}{1}$ $4/1$ by the LCD

	$=$ $=$	$\frac{4 (x + 5)}{x + 5} + \frac{7}{x + 5}$ $(4(x+5))/(x+5) + 7/(x+5)$	Simplify

	$=$ $=$	$\frac{4 (x + 5) + 7}{x + 5}$ $(4(x+5)+7)/(x+5)$	Combine the numerators

	$=$ $=$	$\frac{4 x + 20 + 7}{x + 5}$ $(4x+20+7)/(x+5)$	Distribute $4$ $4$ inside parenthesis

	$=$ $=$	$\frac{4 x + 27}{x + 5}$ $(4x+27)/(x+5)$	Simplify

\frac{4 x + 27}{x + 5}

$(4x+27)/(x+5)$

Question 6 of 7

Perform the operation

3 x^{2} + \frac{x + 2}{x^{2} - 4}

$3x^2+(x+2)/(x^2-4)$

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

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Adding and Subtracting Rational Expressions

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

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

Factorise the denominator of the second term.

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

	$=$ $=$	$3 x^{2} + \frac{1}{x - 2}$ $3x^2+1/(x-2)$	Cancel out $x + 2$ $x+2$

	$=$ $=$	$\frac{3 x^{2}}{1} + \frac{1}{x - 2}$ $(3x^2)/1+1/(x-2)$	Write $3 x^{2}$ $3x^2$ as a fraction

	$=$	$\frac{3x^2 \times \color{red}{(x-2)}}{1 \times \color {red}{(x-2)}} + \frac{1}{x-2}$	Multiply $\frac{3 x^{2}}{1}$ $(3x^2)/1$ by the LCD

	$=$ $=$	$\frac{3 x^{2} (x - 2)}{x - 2} + \frac{1}{x - 2}$ $(3x^2(x-2))/(x-2) + 1/(x-2)$	Simplify

	$=$ $=$	$\frac{3 x^{2} (x - 2) + 1}{x - 2}$ $(3x^2(x-2)+1)/(x-2)$	Combine the numerators

	$=$ $=$	$\frac{3 x^{3} - 6 x^{2} + 1}{x - 2}$ $(3x^3-6x^2+1)/(x-2)$	Distribute $3 x^{2}$ $3x^2$ inside parenthesis

\frac{3 x^{3} - 6 x^{2} + 1}{x - 2}

$(3x^3-6x^2+1)/(x-2)$

Question 7 of 7

Perform the operation

x - 2 + \frac{x + 4}{x - 5}

$x-2+(x+4)/(x-5)$

1. $\frac{x^{2} - 8 x + 14}{x - 5}$ $(x^2-8x+14)/(x-5)$
2. $\frac{x^{2} - 6 x + 14}{x + 5}$ $(x^2-6x+14)/(x+5)$
3. $\frac{x^{2} - 8 x + 14}{x + 5}$ $(x^2-8x+14)/(x+5)$
4. $\frac{x^{2} - 6 x + 14}{x - 5}$ $(x^2-6x+14)/(x-5)$

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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.

$x - 2 + \frac{x + 4}{x - 5}$ $x-2+(x+4)/(x-5)$	$=$ $=$	$\frac{x - 2}{1} + \frac{x + 4}{x - 5}$ $(x-2)/1+(x+4)/(x-5)$	Convert $x - 2$ $x-2$ into a fraction

	$=$	$\frac{(x-2) \times \color{red}{(x-5)}}{1 \times \color {red}{(x-5)}} + \frac{x+4}{x-5}$	Multiply $\frac{x - 2}{1}$ $(x-2)/1$ by the LCD

	$=$ $=$	$\frac{(x - 2) (x - 5)}{x - 5} + \frac{x + 4}{x - 5}$ $((x-2)(x-5))/(x-5) + (x+4)/(x-5)$	Simplify

	$=$ $=$	$\frac{(x - 2) (x - 5) + x + 4}{x - 5}$ $((x-2)(x-5)+x+4)/(x-5)$	Combine the numerators

	$=$ $=$	$\frac{x^{2} - 7 x + 10 + x + 4}{x - 5}$ $(x^2-7x+10+x+4)/(x-5)$	Distribute $x - 2$ $x-2$ inside parenthesis

	$=$ $=$	$\frac{x^{2} - 6 x + 14}{x - 5}$ $(x^2-6x+14)/(x-5)$	Simplify

\frac{x^{2} - 6 x + 14}{x - 5}

$(x^2-6x+14)/(x-5)$

Add and Subtract Algebraic Fractions

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Quiz summary

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Adding and Subtracting Rational Expressions

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Adding and Subtracting Rational Expressions

Quizzes

$\frac{4 x}{14} + \frac{6 x}{14}$ $(4x)/14 +(6x)/14$	$=$ $=$	$\frac{10 x}{14}$ $(10x)/14$

	$=$ $=$	$\frac{5 x}{7}$ $(5x)/7$	Express in lowest terms