Question

Given that x < 5, rewrite 5x - |x - 5| without using absolute value signs.

Answer 1

For $x$, $x-5$, therefore $|x-5|=-(x-5)=-x+5$

So, for $x$, $5x-|x-5|=5x-(-x+5)=5x+x-5=6x-5$

Answer 2

So, if we have an arbitrary number under the absolute value, say y, we have |y| to be:

$y0 \implies |y| =y$

Which in plain english means "If y is less than 0, then the absolute value of y is the negation of y, and if y is greater than 0 then the absolute value of y is y."

So, in our above absolute value expression we would have:

$x-5 < 0 \implies |x-5| = -(x-5) \\ x-5>0 \implies |x-5| = x-5$

Since we are given x<5, we know that the absolute value actually negates what's encompassed by it so we have:

$5x-(-(x-5))=\\ 5x-(-x+5)= 5x+x-5= 6x-5$

So our expression without the absolute value signs is simply 6x-5.