If $a<b$ why is $a \le b$

Question

I recently showed a problem to someone and they stated the following:

If $a<b$ then $a\le b$.

I find this very confusing how can if we stated that $a$ is strictly less than $b$ how can we then say that $a$ is less than $b$ and could be equal to $b$? For example if I wanted to solve an equation where a strict inequality was stated putting $x\le b$ would be incorrect right?

Answer 1

$a\le b$ is an abbreviation for $a<b\lor a=b$ ($\lor$ denoting the inclusive or, which satisfies $p\implies(p\lor q)$).

Answer 2

Define $\leq$ first say on the set of integers:

$a\leq b :\Leftrightarrow \exists c\in\Bbb Z_{\geq 0}:\; a+ c =b$.

Then define $<$ as follows

$a<b :\Leftrightarrow a\leq b \wedge a\ne b.$