Question

The epic gamers who made my lesson didnt even gloss over this

Answer 1

When you have absolute values, that esentially shows how far from zero the number is. This goes both ways — the absolute value of 5 and -5 is the same, as both of them are 5 “places” away from 0.

When solving anything with absolute values, you must do separate “cases”. In this case you have 2 of those — when the expression in the absolute brackets is negative, and when its positive.

First, lets say its positive. The absolute value is 2x - 3, and if its positive it just stays the original way (2x - 3) and you solve it normally:

|2x - 3| - 4 <= 5
2x - 3 - 4 <= 5
2x <= 5 + 3 + 4
2x <= 12
x <= 6

Then we have when its negative, where you put a minus in front of the expression under the brackets:

-(2x - 3) - 4 <= 5
-2x + 3 - 4 <= 5
-2x - 1 <= 5
-2x <= 6 (multiply by -1, and the inequality sign changes)
2x => -6
x => -3

So your x would be -3 <= x <= 6.

Answer 2

So you do whats in the lines first

|2x - 3| = -x

so that leaves you with -x -4 ≤ 5

next you subtract -x from 4

-x - 4 = -5x

that leaves you with -5x ≤ 5

the answer is A)