for what values of x, where a is a positive real number, is |x-a| = |x|-a
why?
To find the values of x that satisfy the given equation, |x-a| = |x| - a, we can consider two cases: when x is greater than or equal to a, and when x is less than a.
Case 1: x ≥ a
In this case, |x-a| simplifies to (x-a), and |x| simplifies to x. So the equation becomes:
x - a = x - a
Here, we see that no matter what value of x we choose, the equation will always be true. So for x ≥ a, all values of x satisfy the equation.
Case 2: x < a
In this case, |x-a| simplifies to (a-x), and |x| simplifies to -x (since x < 0). So the equation becomes:
a - x = -x - a
By simplifying, we get:
a - a = -x - (-x)
0 = 0
Here, we see that the equation is true regardless of the value of x. So for x < a, all values of x satisfy the equation.
Combining the results from Case 1 and Case 2, we can conclude that the equation |x-a| = |x| - a is true for all values of x, where a is a positive real number.