Find the complete range of values of x for which |x+3| < 2x-5.
A) x > 5/2
B) x < 5/2
C) x > 8
D) x > 9
E) x > 2/3[/b]
An alternate approach is to test one value to the right and left of each CRITICAL POINT.
A critical point occurs when the |x+3| = 2x-5.
Case 1: No signs changed
x+3 = 2x-5
8 = x.
To confirm that x=8 is a valid solution, plug it back into the original equation:
|8+3| = 2*8 - 5
11 = 11.
This works.
Thus, x=8 a critical point.
Case 2: Signs changed on one side of the equation
-x-3 = 2x-5
2 = 3x
x = 2/3.
To confirm that x=2/3 is a valid solution, plug it back into the original equation:
|2/3 + 3| = (2)(2/3) - 5
11/3 = -11/3.
Doesn't work.
Thus, x=2/3 is NOT a critical point.
There is only one critical point: x=8.
To determine where |x+3| < 2x-5, test one value to each side of this critical point.
x<8:
Plugging x=0 into |x+3| < 2x-5, we get:
|0+3| < 2*0 - 5
3 < -5.
Doesn't work.
Thus, x<8 is not a valid range.
x>8:
Plugging x=10 into |x+3| < 2x-5, we get:
|10+3| < 2*10 - 5
13 < 15.
This works.
Thus, x>8 is a valid range.
Result:
|x+3| < 2x-5 for all values of x such that x>8.
The correct answer is
C.
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