Solve the inequality for x.

1. Isolate the x terms: To begin, we want to gather all terms containing 'x' on one side of the inequality and all constant terms on the other. Subtract $2x$ from both sides of the inequality:
$$2x - 3 - 2x \geq 4x + 7 - 2x$$
$$ -3 \geq 2x + 7 $$

2. Isolate the constant terms: Next, we want to move the constant term from the right side to the left side. Subtract 7 from both sides of the inequality:
$$ -3 - 7 \geq 2x + 7 - 7 $$
$$ -10 \geq 2x $$

3. Solve for x: Finally, to isolate 'x', divide both sides of the inequality by 2. Since we are dividing by a positive number, the direction of the inequality sign remains the same:
$$ \frac{-10}{2} \geq \frac{2x}{2} $$
$$ -5 \geq x $$

This can also be written as $x \leq -5$.