lx - 2l - lx + 2l > x
The absolute value of x - 2 minus the absolute value of x + 2 greater than x.
What is the Answer Please Help!
x-2 > x + │x+2│ OR -x+2 > x + │x+2│
-2 > │x+2│ or -2x + 2 > │x+2│
-2 > │x+2│ is the same as │x+2│ < -2 which is not true by definition of absolute value.
So -2x + 2 > │x+2│
then │x+2│ < -2x + 2
then x+2 < -2x+2 AND -x-2 < -2x + 2
3x < 0 AND x < 4
so x < 0
Ask yourself this:
1. If x>0, is |x - 2| - |x + 2| positive or negative or zero? (or think of it as "is |x - 2| > |x + 2| ?"
2. If x==0, then clearly |x - 2| - |x + 2| is zero.
3. Now, the interesting one: what if x is negative? Then
is |x - 2| > |x + 2| ?
To solve this inequality, we need to use some rules and properties of absolute value.
First, let's simplify the expression.
We have "lx - 2l - lx + 2l > x".
The absolute value of a number is always non-negative, meaning it is either positive or zero. Therefore, we can simplify the equation further using the fact that |a| - |b| = |a - b|.
So, our equation becomes |x - 2 - (x + 2)| > x.
Simplifying further, we have |x - 2 - x - 2| > x.
Now, simplify the expression inside the absolute value signs. We have |-4| > x.
And this simplifies to 4 > x.
So, the answer to the inequality is x < 4.
In conclusion, to solve the inequality lx - 2l - lx + 2l > x, we simplify it step by step and end up with the solution x < 4.