Question

X-1/x-5<0 Ответ: 1<x<5 Почему? Объясните пожалуйста подробно

Answer 1

Так как частное меньше нуля, числитель или же знаменатель должен быть меньше 0 ,но ни в коем случае не одновременно, ведь - на - =+

Answer 2

To understand why the inequality $\frac{{x-1}}{{x-5}} < 0$ implies that $1 < x < 5$, we need to analyze the critical points and intervals where the expression $\frac{{x-1}}{{x-5}}$ changes its sign.

Step 1: Find critical points The critical points are the values of $x$ that make the expression $\frac{{x-1}}{{x-5}}$ equal to zero or cause the denominator to be zero (which results in an undefined expression). In this case, we set the numerator equal to zero to find the critical point:

$x - 1 = 0$ $x = 1$

So, $x = 1$ is a critical point.

Step 2: Determine sign changes on intervals Now, we want to find the intervals where $\frac{{x-1}}{{x-5}}$ changes its sign. To do this, we'll examine three intervals based on the critical point $x = 1$ and the denominator $x - 5$:

Interval 1: $x < 1$ Choose a value of $x$ less than 1, for example, $x = 0$:

$\frac{{0 - 1}}{{0 - 5}} = \frac{{-1}}{{-5}} = \frac{1}{5} > 0$

So, on this interval, $\frac{{x-1}}{{x-5}} > 0$.

Interval 2: $1 < x < 5$ Choose a value of $x$ between 1 and 5, for example, $x = 2$:

$\frac{{2 - 1}}{{2 - 5}} = \frac{1}{-3} < 0$

So, on this interval, $\frac{{x-1}}{{x-5}} < 0$.

Interval 3: $x > 5$ Choose a value of $x$ greater than 5, for example, $x = 6$:

$\frac{{6 - 1}}{{6 - 5}} = \frac{5}{1} > 0$

So, on this interval, $\frac{{x-1}}{{x-5}} > 0$.

Step 3: Combine the intervals Now that we have identified the sign of $\frac{{x-1}}{{x-5}}$ on each interval, we can conclude that the original inequality $\frac{{x-1}}{{x-5}} < 0$ holds only on Interval 2: $1 < x < 5$. This means that $x$ must be greater than 1 and less than 5 to satisfy the inequality.

Thus, the solution to the inequality is $1 < x < 5$.