To solve the given compound inequality:

$x + 5 \geq 2 \quad \text{and} \quad 5x - 7 \geq 3$

Step 1: Solve each inequality individually.

For $x + 5 \geq 2$:

$x \geq 2 - 5$ $x \geq -3$

For $5x - 7 \geq 3$:

$5x \geq 3 + 7$ $5x \geq 10$ $x \geq \frac{10}{5}$ $x \geq 2$

Step 2: Combine the solutions.

Since it is a compound inequality with "and," we take the intersection of the solutions:

• From $x \geq -3$, the possible values are $x \geq -3$.
• From $x \geq 2$, the possible values are $x \geq 2$.

The overlap (intersection) of $x \geq -3$ and $x \geq 2$ is: $x \geq 2$

Final Solution:

$x \geq 2$

Let me know if you have any questions or want further clarifications.

Related Questions:

1. How would the solution differ if it were "or" instead of "and" in the compound inequality?
2. What are the key differences between solving compound inequalities and absolute value inequalities?
3. How would you graph the solution on a number line?
4. Can inequalities have solutions that are a range rather than a single value?
5. How can inequalities be used to model real-life constraints or problems?

Tip: Always check the direction of the inequality sign and remember to reverse it if you multiply or divide by a negative number!