Precalculus (6th Edition) Blitzer

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

Chapter 1 - Section 1.7 - Combinations of Functions; Composite Functions - Exercise Set - Page 260: 120

Answer

The value of x in the equation $\frac{x-1}{5}-\frac{x-3}{2}=1-\frac{x}{4}$ is $-54$.

Work Step by Step

Consider the equation $\frac{x-1}{5}-\frac{x-3}{2}=1-\frac{x}{4}$ Multiplying both sides by 20 in the original equation we get, $\begin{align} & 20\left( \frac{x-1}{5}-\frac{x-3}{2} \right)=20\left( 1-\frac{x}{4} \right) \\ & 4\left( x-1 \right)-10\left( x-3 \right)=20-5x \end{align}$ The fractions are cleared now; open the brackets and combine the x terms and the constant terms to get $\begin{align} & 4x-4-10x-30=20-5x \\ & 4x-10x+5x-4-30-20=0 \\ & -x-54=0 \end{align}$ Adding $54$ on both sides of the equation we get, $\begin{align} & -x-54+54=54 \\ & -x=54 \end{align}$ Dividing by $-1$ on both sides of the equation we get $\begin{align} & \frac{-x}{-1}=\frac{54}{-1} \\ & x=-54 \end{align}$ Hence, $x=-54$ Substituting $x=-54$ in the original solution we get Left-hand side: $\begin{align} & =\frac{-54-1}{5}-\frac{-54+3}{2} \\ & =\frac{-55}{5}-\frac{-51}{2} \\ & =-11+25.5 \\ & =14.5 \end{align}$ Right-hand side: $\begin{align} & =1-\frac{-54}{4} \\ & =1-\left( -13.5 \right) \\ & =1+13.5 \\ & =14.5 \end{align}$ Hence, the left-hand side is equal to the right-hand side. The value of x is true for the original equation. Hence, the value of x in the equation $\frac{x-1}{5}-\frac{x-3}{2}=1-\frac{x}{4}$ is $-54$.
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