Let's solve the two inequalities step by step.

1st Inequality:

$3 - 4x \geq 1 + 2x$

Steps:

1. Combine like terms: Subtract $2x$ from both sides: $3 - 4x - 2x \geq 1$ This simplifies to: $3 - 6x \geq 1$

2. Subtract 3 from both sides: $-6x \geq -2$

3. Divide through by $-6$, remembering to reverse the inequality sign since dividing by a negative flips the direction: $x \leq \frac{1}{3}$

Solution for the 1st inequality:

$x \leq \frac{1}{3}$

2nd Inequality:

$3(x - 1) \geq -2x - 3$

Steps:

1. Expand the left-hand side: $3x - 3 \geq -2x - 3$

2. Combine like terms: Add $2x$ to both sides: $3x + 2x - 3 \geq -3$ This simplifies to: $5x - 3 \geq -3$

3. Add 3 to both sides: $5x \geq 0$

4. Divide through by 5: $x \geq 0$

Solution for the 2nd inequality:

$x \geq 0$

Combined Solution:

From the two inequalities:

1. $x \leq \frac{1}{3}$
2. $x \geq 0$

The combined solution is the overlap: $0 \leq x \leq \frac{1}{3}$

Do you need a detailed explanation for any part of the process?

Here are some follow-up questions to deepen your understanding:

1. How do you solve compound inequalities with overlapping conditions?
2. What happens when there is no overlap between solutions of two inequalities?
3. Can you graph the solution on a number line to visualize it?
4. How would the solution change if one inequality was strict (e.g., $>$ or $<$) instead of $\geq$ or $\leq$?
5. How can inequalities be used to solve real-world problems like optimization?

Tip: Always reverse the inequality sign when multiplying or dividing by a negative number!