To solve the inequality $-3x + 5 \geq 4x + 5$, follow these steps:

1. Isolate the variable term: Move all terms involving $x$ to one side of the inequality.

$-3x + 5 \geq 4x + 5$

Subtract $4x$ from both sides:

$-3x - 4x + 5 \geq 5$

This simplifies to:

$-7x + 5 \geq 5$

2. Isolate the constant term: Move all constant terms to the other side of the inequality.

Subtract 5 from both sides:

$-7x + 5 - 5 \geq 5 - 5$

This simplifies to:

$-7x \geq 0$

3. Solve for $x$: Divide both sides by $-7$. Remember that dividing by a negative number reverses the inequality sign.

$x \leq \frac{0}{-7}$

Simplifying this:

$x \leq 0$

So, the solution to the inequality is:

$x \leq 0$

Would you like any details or have any questions about the steps involved?

Here are some related questions to further your understanding:

1. What happens to the inequality sign when you divide or multiply both sides by a negative number?
2. How do you solve an inequality when variables are present on both sides?
3. Can you provide an example of solving an inequality with fractions?
4. What are the steps to graphing the solution of an inequality on a number line?
5. How does solving a compound inequality differ from a single inequality?

Tip: Always remember to flip the inequality sign when multiplying or dividing both sides of an inequality by a negative number.