If |x - 3/4| < 9/4, which of the following could be a value of x?

\(\frac{-9}{4}<x-\frac{3}{4}<\frac{9}{4}\)

\(\frac{-6}{4}<x<\frac{12}{4}\)

\(-1.5<x<3\)

Out of all the answer choices, only 2 is within this range

Answer is (C)

In a question like this on Absolute values, instead of using the definition of |x-a|, using the distance approach will help you solve the question much faster.
Remember that |x-a| is the distance of the number x from the number a, on the number line.

So, essentially, this question is telling us that the distance of x from ¾ should be less than 9/4.
Let’s plot a number line and identify the critical points:



The 3 critical points in this situation are:

¾

¾ + \(\frac{9}{4}\) = 3

¾ - \(\frac{9}{4}\) = - 1.5

When we plot these points on the number line, it’s easy to figure out that x has to be between -1.5 and 3 if its distance from ¾ has to be less than \(\frac{9}{4}\).

From this, options A, B and E can be eliminated quite easily. However, option D can be eliminated because of the fact that the distance has to be LESS THAN \(\frac{9}{4}\) and not equal to \(\frac{9}{4}\); hence, x cannot be equal to 3, it has to be LESS THAN 3.

The correct answer option, therefore, is C.

Hope this helps!

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