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vikram4689
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solve for possible value of x ; /x + 1/ + /x - 3/ = 6.
only 3 possible scenarios:
1. when x < -1
(x + 1) is negative in this region and (x - 3) is also negative, so the equation becomes: -(x + 1) - (x - 3) = 6, which simplifies to x = -2
With complex absolute value questions, the solution (x = -2) must be checked against the range for which the equation holds true (x < -1). In this case there is no conflict (since -2 is less than -1) so this is an actual solution.
2. when -1 < x < 3 (x + 1) is positive in this region and (x - 3) is negative, so the equation becomes:(x + 1) - (x - 3) = 6, which simplifies to 4 = 6, i.e. mathematical jibberish!
This means that there is no solution for the equation in this range.
3. when x > 3 (x + 1) is positive in this region and (x - 3) is also positive, so the equation becomes: (x + 1) + (x - 3) = 6, which simplifies to x = 4
This solution (x = 4) check out since it is in the range for which the equation hold true (x > 3).
Therefore, there are two potential solutions to this absolute value equation, x = -2 and 4.
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/x - 4 / > 8
Scenario 1: when x - 4 > 0 (i.e. when x > 4), x - 4 > 8, which simplifies to x > 12.
Scenario 2: when x - 4 < 0 (i.e. when x < 4), -(x - 4) > 8 which simplifies to x < -4
The solution is x > 12 OR x < -4.
*****************************************************************************
Is x > 0 ?
(1) /x - 3/ < 5
(2) /x + 2/ > 5
The first statement can be solved using the method described above.
Scenario 1: When x > 3, x - 3 < 5, which can be simplified to x < 8
Scenario 2: When x < 3, -(x - 3) < 5 or x > -2
Statement (1) can be simplified as -2 < x < 8, which is NOT sufficient to answer the question "is x > 0?"
The second statement can be solved in a similar manner.
Scenario 1: When x > -2, x + 2 > 5 or x > 3.
Scenario 2: When x < -2, -(x + 2) > 5 or x < -7.
Statement (2) can be simplified as x > 3 or x < -7, which is NOT sufficient to answer the question "is x > 0?"
Since both statements are true (remember this is always the case in Data Sufficiency), x must be greater than 3 and less than 8. This is the only overlapping region between the ranges from statements (1) and (2). All of the values between 3 and 8 are positive so TOGETHER the statements are SUFFICIENT and the answer is C.
only 3 possible scenarios:
1. when x < -1
(x + 1) is negative in this region and (x - 3) is also negative, so the equation becomes: -(x + 1) - (x - 3) = 6, which simplifies to x = -2
With complex absolute value questions, the solution (x = -2) must be checked against the range for which the equation holds true (x < -1). In this case there is no conflict (since -2 is less than -1) so this is an actual solution.
2. when -1 < x < 3 (x + 1) is positive in this region and (x - 3) is negative, so the equation becomes:(x + 1) - (x - 3) = 6, which simplifies to 4 = 6, i.e. mathematical jibberish!
This means that there is no solution for the equation in this range.
3. when x > 3 (x + 1) is positive in this region and (x - 3) is also positive, so the equation becomes: (x + 1) + (x - 3) = 6, which simplifies to x = 4
This solution (x = 4) check out since it is in the range for which the equation hold true (x > 3).
Therefore, there are two potential solutions to this absolute value equation, x = -2 and 4.
*****************************************************************************
/x - 4 / > 8
Scenario 1: when x - 4 > 0 (i.e. when x > 4), x - 4 > 8, which simplifies to x > 12.
Scenario 2: when x - 4 < 0 (i.e. when x < 4), -(x - 4) > 8 which simplifies to x < -4
The solution is x > 12 OR x < -4.
*****************************************************************************
Is x > 0 ?
(1) /x - 3/ < 5
(2) /x + 2/ > 5
The first statement can be solved using the method described above.
Scenario 1: When x > 3, x - 3 < 5, which can be simplified to x < 8
Scenario 2: When x < 3, -(x - 3) < 5 or x > -2
Statement (1) can be simplified as -2 < x < 8, which is NOT sufficient to answer the question "is x > 0?"
The second statement can be solved in a similar manner.
Scenario 1: When x > -2, x + 2 > 5 or x > 3.
Scenario 2: When x < -2, -(x + 2) > 5 or x < -7.
Statement (2) can be simplified as x > 3 or x < -7, which is NOT sufficient to answer the question "is x > 0?"
Since both statements are true (remember this is always the case in Data Sufficiency), x must be greater than 3 and less than 8. This is the only overlapping region between the ranges from statements (1) and (2). All of the values between 3 and 8 are positive so TOGETHER the statements are SUFFICIENT and the answer is C.
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