Absolute Value Conumdrum

[guitarguy]

I have the equation:

|4x+7| - |3x+3|= 4

I want to solve for x.

My work:

I take the positive of both expressions.

4x+7-3x-3=4

and I get x=0 which checks.

Then I take the negative of one expression.

-4x-7-3x-3=4

and I get x=-2 which does not check!


Why doesn't it check?

And how would I solve equations with any number of absolute value expressions and a constant?

 

[Quaid]

|4x+7| - |3x+3|= 4

I take the positive of both expressions.

4x+7-3x-3=4

and I get x=0 which checks.

Then I take the negative of one expression.

-4x-7-3x-3=4

and I get x=-2 which does not check!

Why doesn't it check?

Hi guitarguy:

-2 does not check because -2 is not less than -7/4 AND greater than -1, simultaneously.

In other words, when you took the negative of 4x+7, you're implicitly stating that x is less than -7/4. (That jives with x=-2.)

BUT, you used the positive 3x+3, so you're also implicitly stating that x is greater than -1. (That doesn't jive with x=-2.)

Take the negative of both 4x+7 and 3x+3, and you'll obtain the other solution.

If you're still puzzled, draw the Real number line, and figure out the range of x for each absolute-value expression (i.e., take cases).