Answer: (0, 25) ∪ (25, ∞)
Explanation:
The expression is undefined when the denominator is equal to zero and when the radicand (expression inside the radical) is less than zero.
Let's deal with the denominator first by finding the x-values that would make the denominator equal to zero (these are the values that will NOT be included in the interval notation)
x - 5√x = 0
x = 5√x added 5√x to both sides
x² = 25x squared both sides
x² - 25x = 0 subtracted 25x from both sides
x(x - 25) = 0 factored the quadratic equation
x = 0 and x - 25 = 0 applied the Zero Product Property
x = 0 and x = 25 solved for x
So, x ≠ 0 and x ≠ 25
Next, let's deal with the radicand by setting the expressions inside the radicals to "greater than or equal to zero".
2√x and 5√x both have a radicand of x
x ≥ 0
Now, let's put it all together
x ≥ 0 and x ≠ 0 and x ≠ 25
so, x is between 0 and ∞ but not including 0 or 25.
Interval Notation: (0, 25) ∪ (25, ∞)
Reminder that interval notation uses a bracket [ ] when the value is included and a parenthesis ( ) when the value is not included. Infinity always uses a parenthesis since it is not an actual number.