# What should $a$ be for the system of inequalities to have one solution?

What should $$a$$ be for the system of inequalities to have one solution?

$$(1) \frac{3}{x-a}\geq 1 (2)\left | x-2a-2 \right | \leq 1$$

For them to have one solution, the solution must be an endpoint of an interval. (1) gives me $$x\leq a+3$$, but I don't know how to proceed from here. Any hint would be appreciated.

The first it's $$\frac{3}{x-a}-1\geq0$$ or $$\frac{3+a-x}{x-a}\geq0$$ or $$a The second it's $$2a+1\leq x\leq3+2a.$$ Now, we see that an unique possibility it's $$3+a=2a+1.$$ Can you end it now?
• @Aleksandr The first it's the intervals method. By the definition of $|\cdot|$ we have $|x|\leq a$ it's $-a\leq x\leq a$. Thus, the second it's $-1\leq x-2a-2\leq1$ or $2a+1\leq x\leq2a+3.$ – Michael Rozenberg Aug 3 '19 at 4:08
• Why can't $x-a$ be smaller than $0$? – Aleksandr Aug 3 '19 at 4:31