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If |x - 3| + |x + 9| = 14 then what is the difference between maximum and minimum possible absolute values of x?
- a)4
- b)14
- c)6
- d)0
- e)None of these
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If |x - 3| + |x + 9| = 14 then what is the difference between maximum ...
Case I: x ≥ 3
∴ (x - 3) + (x + 9) = 14
∴ x = 4
Case II: 3 ≥ x ≥ -9
∴ (3 - x) + (x + 9) = 14
This case is not possible.
Case III: x ≤ -9
∴ (3 - x) - (x + 9) = 14
∴ x = -10
The difference between the maximum and the minimum absolute values of x = 10 - 4 = 6 Hence, option 3.
∴ (x - 3) + (x + 9) = 14
∴ x = 4
Case II: 3 ≥ x ≥ -9
∴ (3 - x) + (x + 9) = 14
This case is not possible.
Case III: x ≤ -9
∴ (3 - x) - (x + 9) = 14
∴ x = -10
The difference between the maximum and the minimum absolute values of x = 10 - 4 = 6 Hence, option 3.
Most Upvoted Answer
If |x - 3| + |x + 9| = 14 then what is the difference between maximum ...
Given equation: |x - 3| + |x + 9| = 14
To find the difference between the maximum and minimum possible absolute values of x, we need to solve the equation and determine the range of values for x.
1. Solving the equation
We can rewrite the equation using the definition of absolute value:
(x - 3) + (x + 9) = 14 or (x - 3) - (x + 9) = 14
Simplifying the equation, we get:
2x + 6 = 14 or -2x - 12 = 14
For the first equation, solving for x, we have:
2x = 14 - 6
2x = 8
x = 4
For the second equation, solving for x, we have:
-2x = 14 + 12
-2x = 26
x = -13
2. Determining the range of values for x
The equation |x - 3| + |x + 9| = 14 has two solutions: x = 4 and x = -13.
To find the maximum and minimum possible absolute values of x, we need to evaluate |x - 3| and |x + 9| for both values of x.
For x = 4:
|x - 3| = |4 - 3| = 1
|x + 9| = |4 + 9| = 13
For x = -13:
|x - 3| = |-13 - 3| = |-16| = 16
|x + 9| = |-13 + 9| = |-4| = 4
3. Calculating the difference between the maximum and minimum possible absolute values of x
For x = 4, the difference between the maximum and minimum possible absolute values of x is:
13 - 1 = 12
For x = -13, the difference between the maximum and minimum possible absolute values of x is:
16 - 4 = 12
Therefore, the difference between the maximum and minimum possible absolute values of x is 12.
Hence, the correct answer is option C) 6.
To find the difference between the maximum and minimum possible absolute values of x, we need to solve the equation and determine the range of values for x.
1. Solving the equation
We can rewrite the equation using the definition of absolute value:
(x - 3) + (x + 9) = 14 or (x - 3) - (x + 9) = 14
Simplifying the equation, we get:
2x + 6 = 14 or -2x - 12 = 14
For the first equation, solving for x, we have:
2x = 14 - 6
2x = 8
x = 4
For the second equation, solving for x, we have:
-2x = 14 + 12
-2x = 26
x = -13
2. Determining the range of values for x
The equation |x - 3| + |x + 9| = 14 has two solutions: x = 4 and x = -13.
To find the maximum and minimum possible absolute values of x, we need to evaluate |x - 3| and |x + 9| for both values of x.
For x = 4:
|x - 3| = |4 - 3| = 1
|x + 9| = |4 + 9| = 13
For x = -13:
|x - 3| = |-13 - 3| = |-16| = 16
|x + 9| = |-13 + 9| = |-4| = 4
3. Calculating the difference between the maximum and minimum possible absolute values of x
For x = 4, the difference between the maximum and minimum possible absolute values of x is:
13 - 1 = 12
For x = -13, the difference between the maximum and minimum possible absolute values of x is:
16 - 4 = 12
Therefore, the difference between the maximum and minimum possible absolute values of x is 12.
Hence, the correct answer is option C) 6.
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