Write the following expressions without using absolute values.
(1) |b - 3| if ≥ 3
(2) |u - v|/|v - u| if u does not = v, u does not = 0 and v does not = 0.
(3) |a - 5| if a < 5
I like to think about the absolute value of the difference of two numbers as the same thing as the bigger number minus to smaller one. |b-a|=|a-b| but depending on whether a>b or b>a the difference without the absolute value could be positive or negative.
For #1 you are just considering when b is greater than or equal to 3. What do you notice about numbers that are bigger than 3? Try plugging in b=4,5,10 and so on. Do they all have the same sign? If so, does the absolute value change anything?
1) from the definition of absolute value, we get
$\displaystyle \left\vert b-3\right\vert =\left\{
\begin{array}
[c]{cc}%
(b-3), & for\text{ }(b-3)\geq0\text{ or }b\geq3\text{ }\\
(3-b), & for\text{ }(b-3)<0\text{ or }b<3\text{ }%
\end{array}
\right.$
hence
$\displaystyle \left\vert b-3\right\vert =(b-3)$ for $\displaystyle b\geq3\$
I spent too much time just now looking up Latex syntax and reviewing logic rules so I'm posting this even though it probably isn't exactly what you want.
$\displaystyle \left \{ \forall b| b\in \mathbb{R} \text{ and } b \ge 3 \right \} \rightarrow \Big( \Big [ (b-3) \ge 0 \Big] \Leftrightarrow |b-3|=b-3 \Big)$
What do you mean by algebraically? You have to refer to the definition of absolute value at some point.