# How do you simplify abs((-(3-1)+5)div(-2))?

$\frac{3}{2}$

#### Explanation:

The absolute value of a number is always positive, but before we can do the absolute value, we need to simplify what is inside of it. So let's do that:

$\left\mid \left(- \left(3 - 1\right) + 5\right) \div \left(- 2\right) \right\mid$

The first thing I'm seeing is that it's a bit difficult to see what is going on, so I'll add some colour and help things a bit. Specifically, I'm going to colour a term that is within brackets that needs to be done before we can so the division (which is the operation that jumped out at me at first!):

$\left\mid \textcolor{g r e e n}{\left(- \left(3 - 1\right) + 5\right)} \div \left(- 2\right) \right\mid$

So I'm going to do the green section separately, then put the result back in:

$\textcolor{g r e e n}{\left(- \left(3 - 1\right) + 5\right)} = \left(- \left(2\right) + 5\right) = \left(- 2 + 5\right) = \left(3\right)$

and now let's put the solution to this part back into the original:

$\left\mid 3 \div \left(- 2\right) \right\mid = \left\mid - \frac{3}{2} \right\mid$

and now we can take the absolute value:

$\left\mid - \frac{3}{2} \right\mid = \frac{3}{2}$

Jan 14, 2017

$\frac{3}{2}$

#### Explanation:

$\textcolor{b l u e}{\text{Introduction to concept}}$

The | | turns what ever is inside it into a positive value.
AS an example: $| - 7 | = | + 7 | = 7$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Consider: "color(green)((-color(red)((3-1))+5)

Bracket within brackets is called nested brackets. In this situation you consider each pair of brackets working outwards

Step 1: $\text{ } \textcolor{red}{\left(3 - 1\right) = + 2}$

Step 2: " "color(green)((-color(red)((3-1))+5)" "->" "color(green)((-color(red)((+2))+5)

$\text{ "->" } \left(- 2 + 5\right) = \left(+ 3\right)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Putting it all together}}$

$| \textcolor{w h i t e}{.} \left(- \left(3 - 1\right) + 5\right) \div \left(- 2\right) \textcolor{w h i t e}{.} | \text{ " =" } | \textcolor{w h i t e}{.} 3 \div \left(- 2\right) \textcolor{w h i t e}{.} |$

$\text{ "=" } | \textcolor{w h i t e}{.} - \frac{3}{2} \textcolor{w h i t e}{.} |$

$\text{ "=" } + \frac{3}{2}$