MHB How Can Absolute Value Definitions Be Used to Rewrite Expressions?

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The absolute value definition states that |x| equals x for non-negative x and -x for negative x. This can be applied to rewrite expressions lacking absolute values. For example, the expression |x^4 + 1| can be simplified, as x^4 + 1 is always greater than zero. Therefore, |x^4 + 1| equals x^4 + 1 for all real numbers x. This confirms that the expression remains unchanged when applying the absolute value definition in this case.
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The definition of absolute value states the following:

| x | = x when x is > or = 0

| x | = -x when x < 0

I can use the above definition to rewrite expressions that do not contain absolute values.

| x^4 + 1 |

Since x^4 + 1 > 0, then the answer is simply the expression itself or x^4 + 1.

Correct?
 
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Yes, for all real numbers, x, |x^4+ 1|= x^4+ 1.
 

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