Is this a correct way to use triangle inequality

Question

If I have:

$$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq f(x^*)$$

Can I proceed to say:

$$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq |g_1(x) - g_2(x)| - |(g_1(a) - g_2(a))|$$ $$ \implies |g_1(x) - g_2(x)| \leq |g_1(a) - g_2(a)| + f(x^*)$$

Little bit confused because, how can we say the over-estimate is still $\leq f(x^*)$

I feel as if the first line should read:

$$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \geq |g_1(x) - g_2(x)| - |g_1(a) - g_2(a)|$$

This would make more sense, wouldn't it?

Answer 1

$|a-b|\ge|a|-|b|$. So your last inequality works, not the first.