Remember that the absolute value of a number is the distance that number is from 0 on a number line and that it is always positive.
e.g.
|12| = 12 because 12 is 12 away from 0 and
|-12| = 12 as well because -12 is also 12 away from 0
If x = -3 that means that anywhere you see an x in the expression
|x - 4| - x2 you can replace x with -3 because the statement x = -3 means "x is the same thing as -3"
since x is the same thing as -3, I can rewrite
|x - 4| - x2 as
|(-3) - 4| - (-3)2
In this you want to keep track of your negative signs. Do each part separately if this is giving you trouble.
(-3) - 4 means -3 - 4. You can think of subtracting a positive number as telling you to go left on the number line. Start at -3 and go left 4. When you do this you end up on -7.
You can also think of subtraction as "plus the opposite". -3 - 4 is the same thing as -3 + -4 because -4 is the opposite of 4. Whenever you add two numbers that are the same sign you get another number that is the same sign. You know that 3 + 4 is 7, so you know -3 + -4 is -7.
If squaring the -3 is giving you a problem remember that (-3)2 means multiply -3 by itself:
(-3)2 = (-3)(-3) = 9
If you have trouble remembering what happens when you multiply positive and negative numbers, here's a silly metaphor that may help:
the friend of my friend is my friend + + = +
the friend of my enemy is my enemy + - = -
the enemy of my friend is my enemy - + = -
the enemy of my enemy is my friend - - = +
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|x - 4| - x2 start with this
|(-3) - 4| - (-3)2 because x = -3
|-7| - 9 working out each part above
7 - 9 because abs val is distance from 0
-2 you know 9 - 7 = 2, so "backward" subtraction gives you the opposite of 2.
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If the negative numbers are still giving you trouble when you add and subtract them, remember that on a number line, they're a reflection of the positive numbers. We say that numbers "get smaller" when we go left on this number line and that numbers "get bigger" when we go right on it. (So even though 10 is bigger than 7, -10 is smaller than -7.)
There are only two cases when we subtract two numbers that are not the same.
Either we're doing (bigger) - (smaller) which gives us a positive result, or we're doing (smaller) - (bigger) which gives us a negative result.
10 - 4 = 6
4 - 10 = -6
We still get a six, but the sign changes depending on what we're subtracting from what. Also, that six is how far apart those two numbers are. Doing 10 - 4 think: well, 10 and 4 are six apart. The first number is bigger. So 6. Doing 4 - 10 think: well 4 and 10 are six apart. But the first number is smaller, so -6.
Even when you've got double negatives in there, use the 0 on the number line as a benchmark:
Say you're doing
7 - (-2)
Find how far apart 7 and -2 are. Note that 0 is between them. So the distance from -2 to 0 is 2 and the distance from 0 to 7 is 7. That means that -2 and 7 must be 9 apart all together.
Since 7 is bigger than -2 (because 7 is to the right of 2) we get
7 - (-2) = 9
If we had instead
-2 - 7
We still get that -2 and 7 must be 9 apart, but this time the first number in the subtraction is smaller than the second, so the result will be negative.
Since -2 is smaller than 7 we get
-2 - 7 = -9
If we have something like
-10 - (-8)
We think, -10 and -8 are two apart. Also, -10 is smaller than -8 because it's to the left on the number line, so we get a negative answer
-10 - (-8) = -2
If we instead had
-8 - (-10)
We would think -8 and -10 are two apart. Also, -8 is bigger than -10 because it's to the right on the number line, so we get a positive answer:
-8 - (-10) = 2
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Usually knowing what to do when you have positive and negative numbers is a major hangup you have when you're in algebra - so if that wasn't the issue you were having, sorry for going into so much detail.
