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Section 1-6 : Rational Expressions

8. Perform the indicated operation in the following expression.

\[\frac{3}{{x - 4}} + \frac{x}{{2x + 7}}\]

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We first need the least common denominator for this rational expression.

\[{\mbox{lcd : }}\left( {x - 4} \right)\left( {2x + 7} \right)\] Show Step 2

Now multiply each term by an appropriate quantity to get the least common denominator into the denominator of each term.

\[\frac{3}{{x - 4}} + \frac{x}{{2x + 7}} = \frac{{3\left( {2x + 7} \right)}}{{\left( {x - 4} \right)\left( {2x + 7} \right)}} + \frac{{x\left( {x - 4} \right)}}{{\left( {2x + 7} \right)\left( {x - 4} \right)}}\] Show Step 3

All we need to do now is do the addition and simplify the numerator of the result.

\[\frac{3}{{x - 4}} + \frac{x}{{2x + 7}} = \frac{{3\left( {2x + 7} \right) + x\left( {x - 4} \right)}}{{\left( {x - 4} \right)\left( {2x + 7} \right)}} = \frac{{6x + 21 + {x^2} - 4x}}{{\left( {x - 4} \right)\left( {2x + 7} \right)}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{{x^2} + 2x + 21}}{{\left( {x - 4} \right)\left( {2x + 7} \right)}}}}\]