PSAT Math : How to subtract rational expressions with different denominators

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #3 : Rational Expressions

Simplify.

\(\displaystyle \frac{x+5}{x}-\frac{2x-3}{x^{2}}\)

Possible Answers:

\(\displaystyle \frac{x^{2}+7x+3}{x^{2}}\)

\(\displaystyle 3x+3\)

\(\displaystyle 7x+3\)

\(\displaystyle x+6\)

\(\displaystyle \frac{x^{2}+3x+3}{x^{2}}\)

Correct answer:

\(\displaystyle \frac{x^{2}+3x+3}{x^{2}}\)

Explanation:

\(\displaystyle \frac{x+5}{x}-\frac{2x-3}{x^{2}}\)

Determine an LCD (Least Common Denominator) between \(\displaystyle x\) and \(\displaystyle x^{2}\).

LCD = \(\displaystyle x^{2}\)

\(\displaystyle \frac{x(x+5)}{(x)x}-\frac{2x-3}{x^{2}}\)

Multiply the top and bottom of the first rational expression by \(\displaystyle x\), so that the denominator will be \(\displaystyle x^{2}\).

Distribute the \(\displaystyle x\) to \(\displaystyle x+5\).

\(\displaystyle \frac{x^{2}+5x}{x^{2}}-\frac{2x-3}{x^{2}}\)

Now you can subtract because both rational expressions have the same denominators.

\(\displaystyle \frac{x^{2}+5x-2x+3}{x^{2}}\)

Final Answer. 

\(\displaystyle \frac{x^{2}+3x+3}{x^{2}}\)

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