MHB How to Rewrite Absolute Value Expressions Without Absolute Values?

AI Thread Summary
The discussion focuses on rewriting the expression |x + 3| + 4|x + 3| for x < -3 without using absolute values. It establishes that for x < -3, |x + 3| equals -(x + 3). The correct transformation leads to 5|x + 3| being rewritten as -5(x + 3), simplifying to -5x - 15. Participants clarify that combining terms and maintaining the correct operations is essential in the rewriting process. The final expression is presented as -5(x + 3), effectively eliminating the absolute value.
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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?
 
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RTCNTC said:
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:

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

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

$$5|x+3|$$

Let's look at:

$$x<-3$$

Add 3 to both sides:

$$x+3<0$$

And so our expression becomes:

$$5(-(x+3))=-5(x+3)$$

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

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

-x - 3 - 4x - 12

-5x - 15

-5(x + 3)
 
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