My questions: if we have 3 points why do we have 4 conditions? Next, for each of the letters (a,b,c,d) how do we know which equation to use and why is the inequality used? For example, for a) we do x = -8, but then how to we know the equation to use is -(x+3) - (4-x) = -(8+x), and how did the negative sign come in front of the -x+3 and why? I have the same type of questions for b,c,d so maybe an answer to a will help
Plot the 3 points: -8, -3 and 4 on the number line. Now there are 4 distinct regions on the number line:
The part that lies to the left of -8,
the part that lies between -8 and -3,
the part that lies between -3 and 4 and
the part that lies to the right of 4.
In each of these regions, each of the absolute value expressions will behave differently.
You should know the definition of absolute value:
|x| = x if x >= 0
|x| = -x if x < 0
Consider the equation:
|x+3| - |4-x| = |8+x|
What happens in the region which is at the left of -8 say x = -10
x+3 = -10+3 = -7
Since x+3 is negative, |x+3| = -(x + 3)
4-x = 4 - (-10) = 14
Since 4-x is positive, |4 - x| = 4 - x
8 + x = 8 - 10 = -2
Since x+8 is negative, |8+x| = -(8+x)
This is the logic used. Now repeat it for each region.
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