Answer
$[\displaystyle \frac{3}{2},\frac{11}{2})$
Work Step by Step
The idea is to isolate $x$ in the middle.
Adding, subtracting or multiplying/dividing with a positive number preserves order (the inequality symbols remain).
Multiplying/dividing with a negative number inverts the order (the inequality symbols changes).
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$ 3\leq 4x-3 \lt 19 \qquad $ ... add $3$ to all
$ 6\leq 4x \lt 22 \qquad $ ... divide with $4$
$\displaystyle \frac{6}{4}\leq x \lt \frac{22}{4}$
$\displaystyle \frac{3}{2}\leq x \lt \frac{11}{2}$
... left border included, right border excluded
The solution set is $[\displaystyle \frac{3}{2},\frac{11}{2})$ .
