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Add & Subtract Rational Expressions with Unlike Denominators 1Add & Subtract Rational Expressions with Unlike Denominators 1
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Question 1 of 4
1. Question
Perform the operation`(3x)/(x-8)+(5x)/(8-x)`Hint
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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.`x-8` `=` `-(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` to `x-8` `=` `(3x)/(x-8) – (5x)/(x-8)` Simplify `=` `(3x-5x)/(x-8)` Combine the numerators `=` `(-2x)/(x-8)` `(-2x)/(x-8)` -
Question 2 of 4
2. Question
Perform the operation`(13)/(n^2-16)+(-5)/(4-n)`Hint
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Adding and Subtracting Rational Expressions
`a/c +- b/c = (a+-b)/c`Expand the denominator of the first term.`(13)/(n^2-16)+(-5)/(4-n)` `=` `(13)/((n-4)(n+4))+(-5)/(4-n)` `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
3. Question
Perform the operation`(5x)/(8y)+(3x)/(24y^2)`Hint
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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
4. Question
Perform the operation`1/(a-1)-1/a`Hint
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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))`