petrified17 wrote:
If x is a number between -5 and 15, which of following equations represents the range of x?
A) |x| - 5 < 10
B) |x| + 5 < 10
C) |x - 5| < 10
D) |x + 5| < 10
E) |x + 5| < 0
We know: -5 < x < 15
To find the range expressed in an absolute value inequality (or equation) you can use a fairly easy method. If needed, sketch a number line.
1) Find the midpoint,* which is
5, exactly halfway between -5 and 15.
2) Think about "distance from" to set up the expression, LHS
|x| = the distance of x from 0 on the number line
Similarly, |x - 5| is the distance of x from 5, the midpoint. (The point from which distance is measured has "moved" from 0 to 5, the midpoint of THIS range.)
That's the LHS
3) RHS?
Now use 5, midpoint, and 10, distance, as the "distance from": -5 is a distance of 10
from 5, and 15 is a distance of 10
from 5The x here is not inclusive.
The distance of x from 5 cannot be 10 (which would be |x - 5| = 10), but it can be anything up to 10, namely
|x - 5| < 10
ANSWER C
You can check. (In fact, because these numbers are very manageable, if you got really stuck, you could work from the answer choices.)
|x - 5| < 10, remove absolute value bars, check the two cases
Case 1:
x - 5 < 10
x < 15
Case 2:
x - 5 > -10
x > -5
Together: -5 < x < 15 - Correct
ANSWER C
*Midpoint
\(\frac{(-5 + 15)}{2}\) = 5
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