Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter P - Section P.6 - Rational Expressions - Exercise Set - Page 87: 115


The required solution is True

Work Step by Step

We have the given algebraic expression: $\frac{2x-1}{x-7}+\frac{3x-1}{x-7}-\frac{5x-2}{x-7}=0$ We know that for an algebraic expression, a rational expression is an expression which can be expressed in the form $\frac{p}{q}$, where, both $p\ \text{and }q$ are polynomials and the denominator $q\ne 0$. Now, solve the left-hand side of the given algebraic expression. $\frac{2x-1}{x-7}+\frac{3x-1}{x-7}-\frac{5x-2}{x-7}$ Because the denominator of all the fractions is the same, therefore, simply add or subtract the numerators: $\begin{align} & \frac{2x-1}{x-7}+\frac{3x-1}{x-7}-\frac{5x-2}{x-7}=\frac{2x-1+3x-1-\left( 5x-2 \right)}{\left( x-7 \right)} \\ & =\frac{2x-1+3x-1-5x+2}{\left( x-7 \right)} \\ & =\frac{\left( 2x+3x-5x \right)+\left( -1-1+2 \right)}{\left( x-7 \right)} \\ & =\frac{\left( 5x-5x \right)+\left( -2+2 \right)}{\left( x-7 \right)} \end{align}$ And simplify further: $\begin{align} & \frac{2x-1}{x-7}+\frac{3x-1}{x-7}-\frac{5x-2}{x-7}=\frac{\left( 5x-5x \right)+\left( -2+2 \right)}{\left( x-7 \right)} \\ & =\frac{0}{\left( x-7 \right)} \\ & =0 \end{align}$ Thus, $\frac{2x-1}{x-7}+\frac{3x-1}{x-7}-\frac{5x-2}{x-7}=0$. Hence, the given statement is True.
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