Question

Is this true?
| x | + 2 > | x | for all real numbers x.

Answer 1

Looks like it. Try subtracting |x| from both sides, and also try plugging in a few values for x. Try a few like -50, 0, and 50, just to convince yourself.

Answer 2

| x | + 2 > | x |
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| x | is a non-negative number.
So, when you add it to 2, that non-negative number will always be greater than the non-negative number itself.

So, | x | + 2 > | x | always for real numbers x.

Answer 3

This would be true regardless of |x| being a non-negative number (eg. x + 2 > x) - it's just adding the same real number to both sides of an inequality, so it's really just saying 2 > 0, or that any real number plus two is greater than said real number.