solving absolute value equations?

so we have: 4-3|4-x| = 2|4-x|-1

now you can move -3|4-x| to the right side, to get: 4 = 2|4-x|-1+3|4-x|

and then move the -1 to the left side: 4+1 = 2|4-x|+3|4-x|

now add up those terms to get: 5 = 5|4-x|

divide both sides by 5, to get: 1 = |4-x|

now to get rid of the absolute value you need to replace them with a plus or minus, and get two equations: 1 = +\- (4-x)

rewrite as two separate equations: 1 = 4-x and 1 = -(4-x) = -4+x

solving the first equation:

1 = 4-x => x = 3

and the second:

1 = -4+x => x=5

and there's your answer, x is either 3 or 5.

to double check that plug both answers in the original equation:

4-3|4-3| = 2|4-3|-1

4-3|1| = 2|1|-1

4-3 = 2-1

1 = 1 which is correct.

now test x = 5

4-3|4-5| = 2|4-5|-1

4-3|-1| = 2|-1|-1

4-3 = 2-1

1 = 1 which is also correct.

so the most important part, is to know that when you want to get rid of the absolute value signs, you need to replace them with both + and - and evaluate both equations.

An absolute value is going to be well defined in your book with fancy math language and the like. For our purposes, it means the distance the number is away from zero. Because you can't have a negative distance, this number is ALWAYS POSITIVE.

That is the key point...

So the absolute value (sometimes abs()) of a normal number is just that number without its sign. |-42| = 42

| 6 - 12 | = 6

Given x = 25

abs(x - 75) * 2 = abs(-50) * 2 = 50 *2 = 100.

Hope this helps.

BTW absolute value bar is on the same key as the slash, just above the Enter...

4 - 3|4 - x| = 2|4 - x| - 1

Let |4 - x| = y. The equation becomes:

4 - 3y = 2y - 1

5y = 5

y = 1

|4 - x| = 1

Therefore, 4 - x = 1 or -1

4 - x = 1

x = 3

4 - x = -1

x = 5

So, x = 3, 5

Hope this helps.

4-3(4-x) = 2(4-x)-1

4-12+3x=8-2x-1

-8+3x+2x=7

3x=15

x=5