Question

1. which of the following double inequalities represents the inequality \\(|y - 3| - 4 \le -3\\)?

\\(y - 3 \le -1\\) or \\(y - 3 \ge 1\\)

\\(-1 \le y - 3\\) and \\(y - 3 \le 1\\)

\\(y - 3 \le -1\\) and \\(y - 3 \ge 1\\)

\\(-1 \le y - 3\\) or \\(y - 3 \le 1\\)

Explanation:

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<reasoning>

Isolate the absolute value expression

Using the Absolute Value Inequalities knowledge point
\[

$$\begin{aligned} |y - 3| - 4 &\le -3 \\ |y - 3| &\le -3 + 4 \\ |y - 3| &\le 1 \end{aligned}$$

\]

Rewrite as a compound inequality

Using the Absolute Value Inequalities knowledge point
\[

$$\begin{aligned} -1 \le y - 3 \le 1 \end{aligned}$$

\]
This compound inequality can be written as two separate inequalities joined by the word "AND":
\[
-1 \le y - 3 \quad \text{AND} \quad y - 3 \le 1
\]
</reasoning>

<answer>
<mcq-option>(A) \(y - 3 \le -1 \text{ OR } y - 3 \ge 1\)</mcq-option>
<mcq-correct>(B) \(-1 \le y - 3 \text{ AND } y - 3 \le 1\)</mcq-correct>
<mcq-option>(C) \(y - 3 \le -1 \text{ AND } y - 3 \ge 1\)</mcq-option>
<mcq-option>(D) \(-1 \le y - 3 \text{ OR } y - 3 \le 1\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Isolate the absolute value expression

Using the Absolute Value Inequalities knowledge point
\[

$$\begin{aligned} |y - 3| - 4 &\le -3 \\ |y - 3| &\le -3 + 4 \\ |y - 3| &\le 1 \end{aligned}$$

\]

Rewrite as a compound inequality

Using the Absolute Value Inequalities knowledge point
\[

$$\begin{aligned} -1 \le y - 3 \le 1 \end{aligned}$$

\]
This compound inequality can be written as two separate inequalities joined by the word "AND":
\[
-1 \le y - 3 \quad \text{AND} \quad y - 3 \le 1
\]
</reasoning>

<answer>
<mcq-option>(A) \(y - 3 \le -1 \text{ OR } y - 3 \ge 1\)</mcq-option>
<mcq-correct>(B) \(-1 \le y - 3 \text{ AND } y - 3 \le 1\)</mcq-correct>
<mcq-option>(C) \(y - 3 \le -1 \text{ AND } y - 3 \ge 1\)</mcq-option>
<mcq-option>(D) \(-1 \le y - 3 \text{ OR } y - 3 \le 1\)</mcq-option>
</answer>

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