|4x - 3| < 6How many possible integer values are there for x if |4x-3| < 6 ?
A. ONE
B. TWO
C. THREE
D. FOUR
E. FIVE
=> -6 < 4x -3 < 6
=> -3 < 4x < 9
=> -3/4 < x < 9/4
Hence integer values x can take are : 0, 1, 2
Hence Ans C
Absolute Value
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=> -6 < 4x -3 < 6
=> -3 < 4x < 9
=> -3/4 < x < 9/4
Hence integer values x can take are : 0, 1, 2
Hence Ans C
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asamaverick
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|4x-3| < 6 implies -6 < 4x-3 < 6
Taking these individually:
4x - 3 < 6 --------> 4x < 9 --------> x < 9/4 --------> x <= 2 (since x is integer).
4x - 3) > -6 --------> 4x > -3 --------> x > -3/4 --------> x >= 0 (since x is integer).
So combining these two we can say x = 0, 1 or 2. So answer should be C.
Taking these individually:
4x - 3 < 6 --------> 4x < 9 --------> x < 9/4 --------> x <= 2 (since x is integer).
4x - 3) > -6 --------> 4x > -3 --------> x > -3/4 --------> x >= 0 (since x is integer).
So combining these two we can say x = 0, 1 or 2. So answer should be C.
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Patrick_GMATFix
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I really like akdayal's solution above. Below is an alternative:
In word translations, the absolute value of a difference of two terms means the distance between the two terms on the number line: |4x-3| means the distance between 4x and 3 on the number line. Thus we can rephrase the prompt to say "The distance between 3 and 4x is less than 6". On a number line, this means that 4x can be any value that is less than 6 units away from 3. So 4x can be anything between -3 and 9 (these are the boundaries of values 6 units away from 3).
Furthermore, since we're only interested in integers, we only need to consider 4x = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, or 8}. Of these values, the only ones that would result in integer x are:
To practice similar questions, use the Drill Engine to generate timed drills and set topic='Inequalities & Absolute Values' and difficulty='700+'
-Patrick
In word translations, the absolute value of a difference of two terms means the distance between the two terms on the number line: |4x-3| means the distance between 4x and 3 on the number line. Thus we can rephrase the prompt to say "The distance between 3 and 4x is less than 6". On a number line, this means that 4x can be any value that is less than 6 units away from 3. So 4x can be anything between -3 and 9 (these are the boundaries of values 6 units away from 3).
Furthermore, since we're only interested in integers, we only need to consider 4x = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, or 8}. Of these values, the only ones that would result in integer x are:
- 4x=0 -> x=0
- 4x=4 -> x=1
- 4x=8 -> x=2
To practice similar questions, use the Drill Engine to generate timed drills and set topic='Inequalities & Absolute Values' and difficulty='700+'
-Patrick
- Check out my site: GMATFix.com
- To prep my students I use this tool >> (screenshots, video)
- Ask me about tutoring.
