Solve the rational inequality x-1/x-10<0
To solve the rational inequality x - 1 / x - 10 < 0, we can start by finding the critical points, which are the values of x that make the denominator equal to zero.
For this inequality, the denominator x - 10 cannot be equal to zero, so we have the restriction x ≠ 10.
Now let's find the critical points by setting the numerator equal to zero:
x - 1 = 0
x = 1
Therefore, the critical points are x = 1 and x = 10.
We can now create a number line and mark these critical points:
... | ... 1 ... (x ≠ 10) ... 10 ... | ...
Next, we can choose any test point within each interval and check if it satisfies the inequality.
For the interval x < 1, we can choose x = 0 as a test point:
(0 - 1) / (0 - 10) = -1/-10 = 1/10 > 0
For the interval 1 < x < 10, we can choose x = 5 as a test point:
(5 - 1) / (5 - 10) = 4/-5 = -4/5 < 0
For the interval x > 10, we can choose x = 15 as a test point:
(15 - 1) / (15 - 10) = 14/5 > 0
Based on these test points, we see that the inequality is satisfied for x < 1 and x > 10.
Therefore, the solution to the inequality x - 1 / x - 10 < 0 is x < 1 or x > 10.
In interval notation, this can be written as (-∞, 1) U (10, ∞).