Solve the inequality.

Step 1: Solve the inequality.
First, distribute the $\frac{1}{2}$ on the right side of the inequality.

$$7x + 4 < \frac{1}{2}(4x + 3)$$ $$7x + 4 < 2x + \frac{3}{2}$$

Next, subtract $2x$ from both sides of the inequality.

$$7x - 2x + 4 < 2x - 2x + \frac{3}{2}$$ $$5x + 4 < \frac{3}{2}$$

Now, subtract 4 from both sides. To do this, express 4 as a fraction with a denominator of 2, which is $\frac{8}{2}$.

$$5x + 4 - 4 < \frac{3}{2} - 4$$ $$5x < \frac{3}{2} - \frac{8}{2}$$ $$5x < -\frac{5}{2}$$

Finally, divide both sides by 5.

$$x < \frac{-\frac{5}{2}}{5}$$ $$x < -\frac{5}{2} \times \frac{1}{5}$$ $$x < -\frac{1}{2}$$

Step 2: Illustrate the answer on a number line.
The solution $x < -\frac{1}{2}$ means all values of $x$ that are strictly less than $-\frac{1}{2}$. On a number line, this is represented by an open circle at $-\frac{1}{2}$ (to indicate that $-\frac{1}{2}$ is not included in the solution) and an arrow extending to the left from the open circle.

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