Question

X^2-9x+14/x-2≤ 0 Решите систему уравнений

Answer 1

$\frac{x^2-9x+14}{x-2} \leq 0$

$\left \{ {{x^2-9x+14=0} \atop {x-2 \neq 0}} \right.$

x²-9x+14=0
D=81-4*14=$\sqrt{25}$=5

x₁=$\frac{9+5}{2} = \frac{14}{2} =7$

x₁=$\frac{9-5}{2} = \frac{4}{2} =2$

$\left \{ {{x=7} \atop {x \neq 2}} \right.$

(Строим координатную прямую, отмечаем точки и смотрим на каком отрезке она ≤0 )

x∈(-∞;-2) U (2;7]

Answer 2

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Answer 3

Solving the System of Equations

To solve the system of equations x^2 - 9x + 14 / (x - 2) ≤ 0, we need to find the values of x that satisfy the inequality.

Let's break down the problem into two parts:

1. Finding the values of x for which x^2 - 9x + 14 = 0. 2. Determining the intervals where the expression x^2 - 9x + 14 / (x - 2) is less than or equal to zero.

Part 1: Solving the Quadratic Equation

The equation x^2 - 9x + 14 = 0 can be factored as (x - 2)(x - 7) = 0. By setting each factor equal to zero, we find two solutions:

- x - 2 = 0 → x = 2 - x - 7 = 0 → x = 7

Therefore, the solutions to the quadratic equation are x = 2 and x = 7.

Part 2: Determining the Intervals

To determine the intervals where the expression x^2 - 9x + 14 / (x - 2) is less than or equal to zero, we need to consider the sign of the expression for different intervals of x.

We can use a sign chart to analyze the intervals:

| Interval | Expression | |----------|------------| | x < 2 | Positive | | 2 < x < 7| Negative | | x > 7 | Positive |

From the sign chart, we can see that the expression is negative for 2 < x < 7.

Therefore, the solution to the inequality x^2 - 9x + 14 / (x - 2) ≤ 0 is 2 < x < 7.

In summary, the system of equations x^2 - 9x + 14 / (x - 2) ≤ 0 is satisfied when 2 < x < 7.