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If y = |x-2| + |x| - |x+2| where x is an integer, then y can take how many non-zero integral values between -10 and 10, exclusive?
- a)10
- b)11
- c)12
- d)13
- e)14
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If y = |x-2| + |x| - |x+2| where x is an integer, then y can take how ...
Given
- x is an integer
- y = |x-2| + |x| - |x+2|
To Find: Number of non-zero integral values of y such that -10 < y < 10
Approach
- We see that y is an expression in modulus (‘absolute value’ is also known as modulus). The expressions inside the modulus will change the signs at their zero points.
- For example, an expression |x-2| will have different signs for x ≥ 2 and x < 2. So, 2 will be the zero point of the expression |x-2|
- So the expression will change signs at -2, 0 and 2. Hence, there will be 4 ranges in which we need to analyse the values of y:
- Range-I: x < -2
- Range-II: -2 ≤ x ≤ 0
- Range-III: 0 < x ≤ 2
- Range-III: x > 2
- We will evaluate the possible values of y for each range such that -10 < y < 10
Working Out
- Range-I: x < -2
- |x-2| = -(x-2), as for x < -2, (x-2) < 0
- |x| = -x, as for x < -2, x < 0
- |x+2| = -(x+2), as for x < -2, (x+2) < 0
- y = -x + 2 – x + x + 2 = -x + 4. So, possible values of y between -10 and 10 = {7, 8, 9}
- Range-II: -2 ≤ x ≤ 0
- |x-2| = -(x-2), as for -2 ≤ x ≤ 0, (x-2) ≤ 0
- |x| = -x, as for -2 ≤ x ≤ 0, x ≤ 0
- (x+2) = x+2, as for -2 ≤ x ≤ 0, (x+2) ≥ 0
- y = - x + 2 – x – x – 2 = -3x. So, possible values of y = {3, 6}
- We did not consider 0, as y should take only non-zero integral values.
- We did not consider 0, as y should take only non-zero integral values.
- Range-III: 0 < x ≤ 2
- |x-2| = (-x-2), as for 0 < x ≤ 2, (x-2) ≤ 0
- |x| = x, as for 0 < x ≤ 2, x > 0
- |x+2| = x+2, as for 0 < x ≤ 2, (x+2) > 0
- y = - x + 2 + x – x – 2 = -x. So, possible values of y = {-1, -2}
- Range-IV: x > 2
- |x-2| = x – 2, as for x > 2, (x-2) > 0
- |x| = x, as for x > 2, x > 0
- |x+2| = x+2, as for x > 2, (x+2) > 0
- y = x – 2 + x – x -2 = x – 4. So, possible values of y = {-1, 1, 2, ……..9}
- So, possible non-zero integral values of y = {-2, -1, 1, 2, ……9}, i.e. a total of 11 values.
Answer: B
Most Upvoted Answer
If y = |x-2| + |x| - |x+2| where x is an integer, then y can take how ...
Understanding the given equation
- The given equation is y = |x-2| + |x| - |x+2|.
- We need to find the number of non-zero integral values y can take between -10 and 10.
Determining the range of x
- To find the range of x for which y is an integer, we can analyze the equation.
- The absolute value function ensures that the result is non-negative.
- We can break down the equation into different cases based on the signs of x-2, x, and x+2.
Counting the number of integral values
- By considering the different cases for x, we can determine the range of y.
- We find that y can take non-zero integral values between -8 and 6, inclusive.
- Therefore, the number of non-zero integral values y can take between -10 and 10, exclusive, is 11.
Therefore, the correct answer is option 'B' (11).
- The given equation is y = |x-2| + |x| - |x+2|.
- We need to find the number of non-zero integral values y can take between -10 and 10.
Determining the range of x
- To find the range of x for which y is an integer, we can analyze the equation.
- The absolute value function ensures that the result is non-negative.
- We can break down the equation into different cases based on the signs of x-2, x, and x+2.
Counting the number of integral values
- By considering the different cases for x, we can determine the range of y.
- We find that y can take non-zero integral values between -8 and 6, inclusive.
- Therefore, the number of non-zero integral values y can take between -10 and 10, exclusive, is 11.
Therefore, the correct answer is option 'B' (11).
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