The image contains two inequalities:

1. $5x - 2 > -42$
2. $5x - 2 < 43$

These are connected by the word "or," meaning we are looking for the solution to one or both inequalities.

Solving the first inequality:

$5x - 2 > -42$ Add 2 to both sides: $5x > -40$ Now, divide by 5: $x > -8$

Solving the second inequality:

$5x - 2 < 43$ Add 2 to both sides: $5x < 45$ Now, divide by 5: $x < 9$

Final solution:

The inequalities are connected by "or," so the solution is: $x > -8 \quad \text{or} \quad x < 9$

This means any $x$ greater than $-8$ or less than $9$ satisfies the condition.

Would you like further details on this solution, or do you have any questions?

Here are 5 related questions for further exploration:

1. How would the solution change if the inequalities were connected by "and" instead of "or"?
2. How would you graph the solution set of these inequalities?
3. What happens if the inequality signs are reversed (i.e., $5x - 2 \geq -42$ or $5x - 2 \leq 43$)?
4. Can you express the solution set using interval notation?
5. What happens if the constant terms (e.g., -42 and 43) are different?

Tip: When solving inequalities, always pay attention to whether you're adding, subtracting, multiplying, or dividing by a negative number, as this will reverse the inequality sign!