show that Conditional Expectation inequality exists

Question

Let $X$ be a random variable with $E[X] < ∞$. Let A be an event with P(A) > 1- $\delta$. How can I show that this inequality with the conditional expectation of X given A exists:

$E[X|A] \geq E[X] - P(A^{c})$

Answer 1

Let $X=1$ or $-1$ with the same probability. $$E [X]=0, \ \ E [X\mid X=1]=1,\ \ E [X\mid X=-1]=-1.$$ Let $A=\{X=-1\}$ and $\delta=\frac23.$

Now, how could the following inequality hold $$-1\geq0-\frac12?$$