Question) Which of the following inequalities is equivalent to −4 < x < 8?
A. |x - 1| < 7
B. |x + 2| < 6
C. |x + 3| < 5
D. |x - 2| < 6
E. None of the above
Solution: We need to form an inequality with absolute value for the inequality −4 < x < 8
This implies that x will take all values between −4 and 8
The mean of −4 and 8 is 2
Also, the distance of the mean (i.e. 2) from −4 or from 8 is 6
Thus: −4 = (2 − 6) and 8 = (2 + 6)
We know that |x − p | = q implies that the distance of x from p is q units
This results in the values (p − q) and (p + q) for x
Thus, comparing, if we have: |x − 2| = 6, we would get x = 2 − 6 = −4 or x = 2 + 6 = 8
Since we need all the values between −4 and 8 (i.e. we should NOT exceed −4 or 8), we should therefore have:
|x − 2| < 6
Answer D
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IIT Kharagpur, TU Dresden Germany
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