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05 Jun 2024, 01:52
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Which of the following inequalities is equivalent to \(-9 \leq x \leq 3\)?
A. \(x^2 \leq 9\)
B. \(|x-3| \leq 6\)
C. \(|x+3| \leq 6\)
D. \(|x+6| \leq 3\)
E. \(3 \leq |x| \leq 6\)
A. \(x^2 \leq 9\)
B. \(|x-3| \leq 6\)
C. \(|x+3| \leq 6\)
D. \(|x+6| \leq 3\)
E. \(3 \leq |x| \leq 6\)
05 Jun 2024, 01:52
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Official Solution:
Which of the following inequalities is equivalent to \(-9 \leq x \leq 3\)?
A. \(x^2 \leq 9\)
B. \(|x-3| \leq 6\)
C. \(|x+3| \leq 6\)
D. \(|x+6| \leq 3\)
E. \(3 \leq |x| \leq 6\)
Let's check each option:
A. \(x^2 \leq 9\)
The above is equivalent to \(|x| \leq 3\), which is equivalent to \(-3 \leq x \leq 3\).
B. \(|x-3| \leq 6\)
The above is equivalent to \(-6 \leq x-3 \leq 6\), which yields \(-3 \leq x \leq 9\).
C. \(|x+3| \leq 6\)
The above is equivalent to \(-6 \leq x+3 \leq 6\), which yields \(-9 \leq x \leq 3\).
D. \(|x+6| \leq 3\)
The above is equivalent to \(-3 \leq x+6 \leq 3\), which yields \(-9 \leq x \leq -3\).
E. \(3 \leq |x| \leq 6\)
The above is equivalent to \(-6 \leq x \leq -3\) or \(3 \leq x \leq 6\).
Answer: C
Which of the following inequalities is equivalent to \(-9 \leq x \leq 3\)?
A. \(x^2 \leq 9\)
B. \(|x-3| \leq 6\)
C. \(|x+3| \leq 6\)
D. \(|x+6| \leq 3\)
E. \(3 \leq |x| \leq 6\)
Let's check each option:
A. \(x^2 \leq 9\)
The above is equivalent to \(|x| \leq 3\), which is equivalent to \(-3 \leq x \leq 3\).
B. \(|x-3| \leq 6\)
The above is equivalent to \(-6 \leq x-3 \leq 6\), which yields \(-3 \leq x \leq 9\).
C. \(|x+3| \leq 6\)
The above is equivalent to \(-6 \leq x+3 \leq 6\), which yields \(-9 \leq x \leq 3\).
D. \(|x+6| \leq 3\)
The above is equivalent to \(-3 \leq x+6 \leq 3\), which yields \(-9 \leq x \leq -3\).
E. \(3 \leq |x| \leq 6\)
The above is equivalent to \(-6 \leq x \leq -3\) or \(3 \leq x \leq 6\).
Answer: C
