The primary way to approach this question is to test the points. For the point to be contained in the solution set, it must satisfy both inequalities.
\(\colorbox{aqua}{$(-4,2)$}\)
$$ 2 \leq 5(-4)-3 $$
$$ 2 \leq -23 ✖ $$
$$ 2(-4)-4 \geq 12+2 $$
$$ -12 \geq 14 ✖$$
\(\colorbox{aqua}{$(1,1)$}\)
$$ 1 \leq 5(1)-3 $$
$$ 1 \leq 2 \checkmark$$
$$ 2(1)-4 \geq 12+1 $$
$$ -2 \geq 13 ✖$$
\(\colorbox{aqua}{$(3,-10)$}\)
$$ -10 \leq 5(3)-3 $$
$$ -10 \leq 12 \checkmark $$
$$ 2(3)-4 \geq 12+(-10) $$
$$ 2 \geq 2 \checkmark $$
\(\colorbox{aqua}{$(6,2)$}\)
$$ 2 \leq 5(6)-3 $$
$$ 2 \leq 27 \checkmark $$
$$ 2(6)-4 \geq 12+2 $$
$$ 8 \geq 14 ✖$$