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Manipulating Rational Expressions

Manipulating Rational Expressions 1.11 - Solution

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a
By substituting x=8x=8 into the rational expression we can find R(8)R(8).
R(x)=x+7x7R(x)=\dfrac{x+7}{x-7}
R(8)=8+787R({\color{#0000FF}{8}})=\dfrac{{\color{#0000FF}{8}}+7}{{\color{#0000FF}{8}}-7}
R(8)=151R(8)=\dfrac{15}{1}
R(8)=15R(8)=15
For x=8x=8, the rational expression has the value 1515.
b
The rational expression is undefined when the denominator equals zero, since division by zero is not allowed. The denominator is x7x-7, so the xx-value that makes the denominator become equal 00 must be 77. We can justify this algebraically by setting the denominator equal to 00 and solve for xx.
x7=0x-7=0
x=7x=7
c
Multiply both the numerator and the denominator by 77.
R(x)=x+7x7R(x)=\dfrac{x+7}{x-7}
R(x)=(x+7)7(x7)7R(x)=\dfrac{(x+7)\cdot 7}{(x-7)\cdot 7}
R(x)=x7+77x777R(x)=\dfrac{x\cdot 7+7\cdot 7}{x\cdot 7-7\cdot 7}
R(x)=7x+497x49R(x)=\dfrac{7x+49}{7x-49}