Hi ziyuenlau,
We're told that X must be an INTEGER, so since we're adding the sums of 3 absolute values - AND we're looking for sums that are LESS than 10 - there cannot be an infinite number of possibilities. Thus, based on the answer choices, there can't be more than 5 possible integer values for N. As such, we can use a bit of 'brute force' to find all of the possible values without too much trouble.
We're given the inequality: |2X - 5| + |X + 1| + |X| < 10.
Let's start with...
X = 0.... |-5| + |1| + |0| = 6 so X could be 0
Now let's work our way "up"...
X = 1.... |-3| + |2| + |1| = 6 so X could be 1
X = 2.... |-1| + |3| + |2| = 6 so X could be 2
X = 3.... |1| + |4| + |3| = 8 so X could be 3
X = 4.... |3| + |5| + |3| = 11 so X CANNOT be 4
Now let's work our way "down"....
X = -1.... |-7| + |0| + |-1| = 8 so X could be -1
At this point, we have 5 possible values of X; since that it the largest of the possible values, we can stop working.
Final Answer:
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