Determine whether x is positive or negative in the following equation:
x-|x|=0
test for a negative:
let x = -5
Left side = -5 - |-5
= -5-5 = -10
≠ RS
test for x = 0,
LS = RS
test for postive x
let x = 5
LS = 5 - |5| = 0 = RS
so x ≥ 0
Recall the definition of |x|:
|x| = x if x >= 0
|x| = -x if x < 0
Now, your equation is
x-|x|=0
x = |x|
so, x >= 0
To determine whether x is positive or negative in the equation x - |x| = 0, you need to analyze two cases: when x is positive and when x is negative.
First, let's assume x is positive:
If x is positive, then |x| will also be positive, since the absolute value of any positive number is positive. Therefore, x - |x| will be positive minus a positive, resulting in a positive value. In this case, x cannot be the solution to the equation x - |x| = 0.
Now, let's assume x is negative:
If x is negative, then |x| will be equal to the positive value of x, since the absolute value of any negative number is its positive equivalent. Therefore, x - |x| can be calculated as x - (-x), which simplifies to x + x, or 2x. In this case, the equation becomes 2x = 0. To solve for x, divide both sides of the equation by 2:
2x/2 = 0/2
x = 0
Hence, x = 0 is the solution to the equation x - |x| = 0 when x is negative.
Therefore, x can be either positive (no solution) or zero (when x is negative).