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The question in the image asks for the solution set of the inequality \(1 - 2x \geq 3\), with multiple choices provided.

### Solving the inequality:
Given:
\[
1 - 2x \geq 3
\]

1. Subtract 1 from both sides:
\[
-2x \geq 3 - 1
\]
\[
-2x \geq 2
\]

2. Divide both sides by \(-2\). Since we're dividing by a negative number, we must reverse the inequality:
\[
x \leq \frac{2}{-2}
\]
\[
x \leq -1
\]

Thus, the solution is \(x \leq -1\).

### Analyzing the options:
- **a.** \([1, \infty)\) — Incorrect, since the solution is for values less than or equal to \(-1\).
- **b.** \(\{x \in \mathbb{R} \mid x \leq -1\}\) — This is correct, as it represents the solution set \(x \leq -1\).
- **c.** \(\{x \in \mathbb{R} \mid x \geq 1\}\) — Incorrect, since the solution is \(x \leq -1\).
- **d.** \((-\infty, 1]\) — Incorrect, as this set includes values greater than \(-1\).
- **e.** \((-\infty, -1]\) — This option is also correct, as it represents the same set as \(x \leq -1\).

Thus, the correct answers are **b** and **e**.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. What happens to the inequality when you divide both sides by a positive number?
2. How do we solve compound inequalities involving both "greater than" and "less than"?
3. What is the graphical representation of the solution set \(x \leq -1\)?
4. How do we interpret inequalities on a number line?
5. How does the solution change if the inequality is \(1 - 2x < 3\) instead of \( \geq \)?

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

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Learn how to solve the inequality 1 - 2x ≥ 3 with a step-by-step explanation. This covers the solution process, including reversing the inequality when dividing by a negative number. Suitable for high school students.

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