The problem consists of two inequalities:

1. $42 < -4x + 2$
2. $-38 \geq -4x + 2$

Solving the first inequality:

$42 < -4x + 2$ Subtract 2 from both sides: $42 - 2 < -4x$ $40 < -4x$ Now, divide both sides by -4, and remember to reverse the inequality sign when dividing by a negative number: $\frac{40}{-4} > x$ $-10 > x$ Or equivalently: $x < -10$

Solving the second inequality:

$-38 \geq -4x + 2$ Subtract 2 from both sides: $-38 - 2 \geq -4x$ $-40 \geq -4x$ Now divide both sides by -4, again reversing the inequality: $\frac{-40}{-4} \leq x$ $10 \leq x$

Combining the results:

We have: $x < -10 \quad \text{or} \quad x \geq 10$

Thus, the solution is $x < -10$ or $x \geq 10$.

Graphing the solution on a number line:

• Draw an open circle at $x = -10$ (because it is a strict inequality $<$).
• Shade to the left of $-10$ (since $x < -10$).
• Draw a closed circle at $x = 10$ (since $x \geq 10$).
• Shade to the right of $10$.

Would you like a detailed graph or further explanation on this?

Here are 5 relative questions:

1. How do you handle inequalities involving multiplication or division by a negative number?
2. Can you combine inequalities into a single inequality when using "or" or "and"?
3. How do strict inequalities differ from non-strict ones in graphing?
4. What are the implications of no solution in inequalities?
5. How can we check if our solution is correct?

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