How to Rewrite Absolute Value Expressions Without Absolute Values?

[mathdad]

The | x | = x when x > or = 0.

The | x | = - x when x < 0.

Rewrite the following expression in a form that does not contain absolute value.

| x + 3 | + 4 | x + 3 |, where x < -3

-(x + 3) + 4 -(x + 3)

-x - 3 + 4 - x - 3

-2x - 6 + 4

-2x - 2

Correct?

 

[MarkFL]

The | x | = x when x > or = 0.

The | x | = - x when x < 0.

Rewrite the following expression in a form that does not contain absolute value.

| x + 3 | + 4 | x + 3 |, where x < -3

-(x + 3) + 4 -(x + 3)

-x - 3 + 4 - x - 3

-2x - 6 + 4

-2x - 2

Correct?

No, you've turned multiplication into addition...we are given the expression:

\(\displaystyle |x+3|+4|x+3|\) where \(\displaystyle x<-3\)

Now, the first thing I would do is combine like terms:

\(\displaystyle 5|x+3|\)

Let's look at:

\(\displaystyle x<-3\)

Add 3 to both sides:

\(\displaystyle x+3<0\)

And so our expression becomes:

\(\displaystyle 5(-(x+3))=-5(x+3)\)

We can stop here because we have rewritten the expression in a form not involving absolute value, and there's no need to distribute in my opinion. :D

 

[mathdad]

|x + 3 | + 4 | x + 3 |, where x < -3

-(x + 3) - 4(x + 3)

-x - 3 - 4x - 12

-5x - 15

-5(x + 3)