Quick help with an equation: |x-3| + |x^2-9| = 0

[Phoenix21]

|x-3| + |x^2-9| = 0

 

[stapel]

|x-3| + |x^2-9| = 0

When is x - 3 > 0? When is it not?

When is x² - 9 > 0? When is it not?

Given these intervals of positivity and negativity, split this equation into cases. For each case, use an interval, and apply the appropriate rule for removing the absolute-value bars. (here). Solve the resulting quadratic, and check to be sure that the solution is within the valid interval.

If you get stuck, please reply showing your efforts so far. Thank you!

 

[pka]

|x-3| + |x^2-9| = 0

This is purely an THOUGHT question.
There it takes a minute to see the solution.
The absolute value is NEVER NEGATIVE.
The only way \(\displaystyle |x-3| + |x^2-9|=0\) can be true is if both summands are zero. \(\displaystyle x=~?\).

 

[Deleted member 4993]

|x-3| + |x^2-9| = 0

Hint:

When two positive numbers added together becomes zero - then each of those, individually, must be equal to zero.

 

[pka]

Hint:

When two positive numbers added together becomes zero - then each of those, individually, must be equal to zero.

Zero is not positive.

 

[Deleted member 4993]

Zero is not positive.

Correction noted - I should have said non-negative (too long to type).