Question

Express the quantity without using absolute value

$a+b+\left|a-b\right|,\mathrm{where}a<b$

Short answer

The quantity of$a+b+\left|a-b\right|\mathrm{is}2b$

Step-by-step solution

Step 1. Apply the concept of absolute value.

If a is a real number, then the absolute value of a is given by:

$\left|a\right|=\left\{\begin{array}{l}a\text{;if}a\ge 0\\ -a\text{;if}a<0\end{array}\right.$

Step 2. Analyze the given value.

Given $a+b+\left|a-b\right|,\text{where}a<b$

Here the value inside modulus is $a-b$.

$\begin{array}{l}a<b\\ \Rightarrow a-b<0\end{array}$

Step 3. Write without using absolute value.

$\begin{array}{l}\left|a-b\right|=-(a-b)\mathrm{.................}[a-b<0]\\ \Rightarrow \left|a-b\right|=-\left(-\left[b-a\right]\right)\\ \Rightarrow \left|a-b\right|=b-a\end{array}$

So,

$\begin{array}{l}a+b+\left|a-b\right|=a+b+b-a\\ \Rightarrow a+b+\left|a-b\right|=2b\end{array}$