Flexxice wrote:
Attachment:
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On the number line, the shaded interval is the graph of which of the following inequalities?
A) \(|x| \leq 4\)
B) \(|x| \leq 8\)
C) \(|x - 2| \leq 4\)
D) \(|x - 2| \leq 6\)
E) \(|x + 2| \leq 6\)
Solution:
Recall that |x - c| = d (where d is positive) means x is exactly d units from c. Therefore, |x - c| ≤ d (where d is positive) means all the values of x that are d units or less from c. For example, let’s analyze choice A, which can be expressed as |x - 0| ≤ 4. If this is the correct answer, then the solution set consists of all the numbers that are 4 units or less from 0, which means the darkened interval on the number line should be from -4 to 4, inclusive. However, that is not what the given graph shows. So A is not the correct answer.
Let’s analyze choice E, which can be expressed as |x - (-2)| ≤ 6. If this is the correct answer, then the solution set consists of all the numbers that are 6 units or less from -2, which means the darkened interval on the number line should be from -8 to 4, inclusive. Since this is exactly what the given graph shows, then E is the correct answer.
(Note: A shortcut to solve this problem using the given graph (and provided that we know the meaning of |x - c| ≤ d) is: The value of c is the midpoint of the two endpoints of the interval and d is the distance between one of the endpoints and c. For example, here the two endpoints of the interval are -8 and 4, so c = (-8 + 4)/2 = -2 and d = |4 - (-2)| = |-8 - (-2)| = 6. Therefore, the correct inequality is: |x - (-2)| ≤ 6, i.e., |x + 2| ≤ 6.)
Answer: E _________________
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