Final answer:
To solve the equation sqrt((5-x)^2) - sqrt((x+5)^2) = 10, we first simplify by using the properties of square roots. After simplifying and considering different cases, we find that x = 5 is the only value that satisfies the equation.
Step-by-step explanation:
To solve the equation sqrt((5-x)^2) - sqrt((x+5)^2) = 10, we first simplify using the properties of square roots. The square root of a squared number is equal to the absolute value of that number. So, we can rewrite the equation as |5-x| - |x+5| = 10.
Next, we consider different cases for the values of x. When x is greater than or equal to -5, both expressions in absolute values are positive. So, we can write |5-x| as 5-x and |x+5| as x+5. When x is less than -5, both expressions in absolute values are negative, so we change the signs: |5-x| becomes -(5-x) = x-5 and |x+5| becomes -(x+5) = -x-5.
Now we can solve for x in these cases:
Case 1: When x is greater than or equal to -5:
5-x - (x+5) = 10
5-x - x-5 = 10
-2x = 10
x = -5
No solutions satisfy this case.
Case 2: When x is less than -5:
(x-5) - (-x-5) = 10
x - 5 + x + 5 = 10
2x = 10
x = 5
Therefore, the only value of x that satisfies the equation is x = 5.