To solve this inequality, we will need to find the critical points by setting the expression equal to zero and then determining the intervals over which the expression is less than zero.
Let's first rewrite the expression in terms of x^1/4 for easier analysis:
x^1/2 - 7x^1/4 + 6 < 0
Now, let's substitute u = x^1/4, which means u^2 = x^1/2:
u^2 - 7u + 6 < 0 (u - 1)(u - 6) < 0
The critical points are u = 1 and u = 6. We need to determine the intervals where the expression is less than zero. This can be done by testing values within each interval.
When u < 1: Choose u = 0: (0 - 1)(0 - 6) = 6 > 0, so exclude this interval.
When 1 < u < 6: Choose u = 3: (3 - 1)(3 - 6) = 2 * (-3) < 0, so include this interval.
When u > 6: Choose u = 7: (7 - 1)(7 - 6) = 6 > 0, so exclude this interval.
Therefore, the solution to the inequality x^1/2 - 7x^1/4 + 6 < 0 is:
To solve this inequality, we will need to find the critical points by setting the expression equal to zero and then determining the intervals over which the expression is less than zero.
Let's first rewrite the expression in terms of x^1/4 for easier analysis:
x^1/2 - 7x^1/4 + 6 < 0
Now, let's substitute u = x^1/4, which means u^2 = x^1/2:
u^2 - 7u + 6 < 0
(u - 1)(u - 6) < 0
The critical points are u = 1 and u = 6. We need to determine the intervals where the expression is less than zero. This can be done by testing values within each interval.
When u < 1:
Choose u = 0: (0 - 1)(0 - 6) = 6 > 0, so exclude this interval.
When 1 < u < 6:
Choose u = 3: (3 - 1)(3 - 6) = 2 * (-3) < 0, so include this interval.
When u > 6:
Choose u = 7: (7 - 1)(7 - 6) = 6 > 0, so exclude this interval.
Therefore, the solution to the inequality x^1/2 - 7x^1/4 + 6 < 0 is:
1 < x^1/4 < 6
or, written in terms of x:
1 < x^(1/4) < 6
This is the solution to the given inequality.