Question 203523
One of the earliest lessons you learn about absolute value is that the absolute value of a positive number is that number. This is also true of the absolute value of 0. And finally the absolute value of a negative number is that number without its minus sign. Another way to express this is: The absolute value of a negative number is the negative of that negative number. Expressing these ideas in mathematical notation:
if {{{a >= 0}}}, then {{{abs(a) = a}}}
if {{{a <  0}}}, then {{{abs(a) = -a}}}
We will use these to simplify: {{{(x-5)^3/abs(x-5)}}}<br>
Case: {{{x-5 >= 0}}}
If {{{x-5 >= 0}}}, then {{{abs(x-5) = x-5}}} so your expression becomes:
{{{(x-5)^3/(x-5)}}}
This simplifies to
{{{(x-5)^2 = (x-5)(x-5)}}}
Substituting {{{abs(x-5)}}} in for {{{(x-5)}}} (Remember they are equal if {{{x-5 >= 0}}} which is true in this case):
{{{(x-5)*abs(x-5)}}}<br>
Case: {{{x - 5 < 0}}}
If {{{x-5 < 0}}}, then {{{abs(x-5) = -(x-5)}}} so your expression becomes:
{{{(x-5)^3/(-(x-5))}}}
This simplifies to
{{{-(x-5)^2 = -(x-5)(x-5)}}}
Substituting {{{abs(x-5)}}} in for {{{-(x-5)}}} (Remember they are equal if {{{x-5 < 0}}} which is true in this case):
{{{abs(x-5)*(x-5) = (x-5)abs(x-5)}}}<br>
In either case we end up with {{{(x-5)abs(x-5)}}}.