Question

1/(x-5)*(x-1)>0 Ответ: x<1 или x>5 Почему?Объясните пожалуйста

Answer 1

Если X больше 1,но меньше 5 то произведение будет отрицательным,

Пример: x=2

1/(2-5)=1/-3=-1/3

2-1=1

-1/3*(1)=-1/3-меньше нуля, не удовлетворяет неравенству

Если X будет меньше 1

Например x=-2

1/(-2-5)=-1/7

-1/7*-3=3/7-удовлетворяет неравенству

Если x>5

Например x=6

1*5=5

5>0-удовлетворяет неравенству

Answer 2

To find the values of x that satisfy the inequality $\frac{1}{(x-5)(x-1)}>0$, we can follow these steps:

Step 1: Find critical points Critical points are the values of x where the expression in the denominator becomes zero, leading to potential changes in the sign of the whole expression. In this case, we need to find the critical points by setting the denominator equal to zero and solving for x:

$x - 5 = 0$ => $x = 5$

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

Step 2: Create test intervals We now divide the number line into intervals based on the critical points we found in step 1. These intervals are: $(-\infty, 1)$, $(1, 5)$, and $(5, \infty)$.

Step 3: Test intervals Now, we will pick test points from each interval and check the sign of the expression $\frac{1}{(x-5)(x-1)}$.

For $(-\infty, 1)$, let's pick $x = 0$ as a test point: $\frac{1}{(0-5)(0-1)} = \frac{1}{5} > 0$ (The expression is positive)

For $(1, 5)$, let's pick $x = 3$ as a test point: $\frac{1}{(3-5)(3-1)} = \frac{1}{(-2)(2)} = \frac{-1}{4} < 0$ (The expression is negative)

For $(5, \infty)$, let's pick $x = 6$ as a test point: $\frac{1}{(6-5)(6-1)} = \frac{1}{5} > 0$ (The expression is positive)

Step 4: Analyze the results Now, based on the signs of the expression in each interval, we can determine the values of x that satisfy the inequality.

• In the interval $(-\infty, 1)$, the expression is positive, so $x < 1$ satisfies the inequality.
• In the interval $(1, 5)$, the expression is negative, so no values in this interval satisfy the inequality.
• In the interval $(5, \infty)$, the expression is positive, so $x > 5$ satisfies the inequality.

Step 5: Final answer The values of x that satisfy the inequality $\frac{1}{(x-5)(x-1)}>0$ are $x < 1$ or $x > 5$.