Chapter A, Problem 8E

To determine

To rewrite: The expression without the absolute value symbol.

Answer to Problem 8E

The equivalent form of the given expression is $\{\begin{array}{l}\left(2x-1\right)\text{ if }x\ge \frac{1}{2}\\ -\left(2x-1\right)\text{ if }x<\frac{1}{2}\end{array}$.

Explanation of Solution

Given:

The expression is $\left|2x-1\right|$.

Definition used:

The absolute value of a number a, denoted by $\left|a\right|$, is the distance from a to 0 on the real

number line. Distances are always positive or 0, so $\left|a\right|\ge 0$ and $\{\begin{array}{l}\left|a\right|=a\text{ if }a\ge 0\\ \left|a\right|=-a\text{ if }a<0\end{array}$.

Calculation:

In the definition of absolute value, substitute $a=2x-1$.

$\left|2x-1\right|=\{\begin{array}{l}\left(2x-1\right)\text{ if }2x-1\ge 0\\ -\left(2x-1\right)\text{ if }2x-1<0\end{array}$

Solve the above expression as,

$\begin{array}{l}2x-1=0\\ 2x=1\\ x=\frac{1}{2}\end{array}$

Thus, the expression becomes,

$\left|2x-1\right|=\{\begin{array}{l}\left(2x-1\right)\text{ if }x\ge \frac{1}{2}\\ -\left(2x-1\right)\text{ if }x<\frac{1}{2}\end{array}$

Hence, the equivalent form of the given expression is $\{\begin{array}{l}\left(2x-1\right)\text{ if }x\ge \frac{1}{2}\\ -\left(2x-1\right)\text{ if }x<\frac{1}{2}\end{array}$.