Question

(x-2)*(x-9)/4x-5 >= 0

Answer 1

ОДЗ
4х-5 неравно 0
Значит х не должно равняться 1,25
1)х-2=0; х=2
2)х-9=0; х=9
(-бескон;1,25)и(1,25;2]и[9;+бескон)

Answer 2

To solve the inequality $\frac{{(x - 2)(x - 9)}}{{4x - 5}} \geq 0$, we need to find the values of $x$ that satisfy the inequality.

First, let's determine the critical points by setting the numerator and denominator equal to zero and finding the values of $x$ that make the expression undefined:

Numerator: $x - 2 = 0$ or $x - 9 = 0$ Solving these equations, we find $x = 2$ and $x = 9$ as critical points.

Denominator: $4x - 5 = 0$ Solving this equation, we find $x = \frac{5}{4}$. However, note that this value does not make the expression undefined, so it's not a critical point.

Now, we'll create a sign chart to analyze the intervals between and around the critical points:

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Interval     |   x - 2   |   x - 9   | 4x - 5  |   Result
------------------------------------------------------
(-∞, 2)      |    -      |    -      |   -     |    +
(2, 9)       |    +      |    -      |   +     |    -
(9, ∞)       |    +      |    +      |   +     |    +

Using the sign chart, we see that the expression is positive or zero in the intervals (-∞, 2) and (9, ∞). It is negative in the interval (2, 9).

Finally, we need to determine the intervals that satisfy the inequality $\frac{{(x - 2)(x - 9)}}{{4x - 5}} \geq 0$. These intervals are where the expression is either greater than or equal to zero.

Therefore, the solution to the inequality is: $x \in (-\infty, 2] \cup [9, +\infty)$