bionication wrote:
Hi guys,
I have a question that involves "opening" an absolute value:
Hypothetical question: If \(|x| > -8\),
logic dictates that x must be lower than -8 or larger than 8. But when I do the actual calculation, I get the opposite result:
\(|x| > -8\)
\(x > -8 and x < -(-8)\)
\(-8 < x < 8\)Can someone comment on what I'm doing incorrectly?
dabral?
Dear
bionicationThe red part in your post is erroneous. And so is the blue part.
Here's how you should think through an inequality like: \(|x| > -8\)
1. The Visual Method|x| represents the distance of a number x from the point 0 on the number line.
This means that the inequality |x| > - 8 represents those numbers that are at a distance greater than -8 units from 0 on the number line.
Now, note that 'distance' is always non-negative. For example, the points -3 and +3 are both said to be at a distance of 3 units from the point 0 on the number line.
So, what is the minimum possible distance between 2 points on the number line? The answer is: ZERO (when 2 points coincide on the number line).
So, what are the numbers that are at a distance greater than -8 units from 0?
The answer is: ALL NUMBERS are at a distance greater than -8 units from 0 (because the distance of any number from 0 will be greater than or equal to 0)
So,
Negative Infinity < x < Positive Infinity is the correct answer for |x| > -8
2. The Algebraic MethodThe given inequality is |x| > - 8
Case 1: x is non-negativeSo, |x| = x
Thus, given inequality becomes: x > - 8 . . . (1)
But for what values of x is x non-negative in the first place? For x > = 0 . . . (2)
So, the values of x that satisfy Case 1 are obtained by the overlap of inequalities (1) and (2), which is x > = 0
So,
Range of x that satisfies Case 1: x > = 0Case 2: x is negativeSo, |x| = -x
Thus, given inequality becomes: -x > - 8
Multiplying both sides of the inequality with -1 will flip the sign of inequality:
x < 8 . . . (1)
But for what values of x is x negative in the first place? For x < 0 . . . (2)
So, the values of x that satisfy Case 2 are obtained by the overlap of inequalities (1) and (2), which is x < 0
So,
Range of x that satisfies Case 2: x < 0(Note: you forgot to consider Inequality (2) in both cases 1 and 2. This is why, you got only x > -8 from Case 1 and likewise, only x < 8 from Case 2. I hope you now realize how important it is to also consider the set of values of x for which a particular case holds true in the first place
)
Thus, we see that the given inequality |x| > -8 is satisfied by x > = 0 or by x < 0. Basically, this inequality is satisfied by ALL POSSIBLE REAL values of x from Negative Infinity to Positive Infinity.
I hope this explanation was useful for you!
Best Regards
Japinder
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