We are tasked with solving the inequality:

$3 - x < 4x + 7$

Step 1: Move all terms involving $x$ to one side of the inequality.

Add $x$ to both sides to eliminate the $-x$ on the left side:

$3 < 5x + 7$

Step 2: Move constant terms to the other side.

Subtract 7 from both sides:

$3 - 7 < 5x$

$-4 < 5x$

Step 3: Solve for $x$.

Divide both sides by 5 to isolate $x$:

$\frac{-4}{5} < x$

This simplifies to:

$x > -\frac{4}{5}$

Final Answer:

The solution is:

$x > -\frac{4}{5}$

This means $x$ must be greater than $-\frac{4}{5}$ for the inequality to hold.

Would you like more details or have any further questions?

Here are some related questions:

1. How do you graph the solution to this inequality on a number line?
2. Can we express this inequality in interval notation?
3. What happens if the inequality sign is reversed, like $3 - x > 4x + 7$?
4. How would you solve a compound inequality that includes this one?
5. What if the coefficients were fractions instead of integers?

Tip: When solving inequalities, remember that dividing or multiplying by a negative number flips the inequality sign!