How to you prove $|a|=|-a|$ is true

Question

Let $a \in R$ prove that:

$|a|=|-a|$

I am new to proofs so this is my attempt:

Case 1: $|a|=|-a|$

$$(a)=-(-a)$$

$$a=a$$

Case 2: $|-a|=|-a|$

$$-(-a)=-(-a)$$

$$a=a$$

Case 3: $|a|=|a|$

$$a=a$$

Is this the correct way to approach a proof like this?

Answer 1

The definition of absolute value of a real number $a$ is $|a|=\sqrt{x^2}$. Since $a^2=(-a)^2$ and square roots only take positive values $|a|=|-a|$

Answer 2

Because the distances from $a$ and from $-a$ to the zero are equal.

Answer 3

If you are going to do a case by case proof, you only need to consider two cases:

Case 1: $a \ge 0 \\ \Rightarrow |a|=a, |-a|=a \\ \Rightarrow |a|=|-a|$

Case 2: $a < 0 \\ \Rightarrow |a|=-a, |-a|=-a \\ \Rightarrow |a|=|-a|$