Question

(5x-10)/(x+8)(x-7)>0

Answer 1

Решение
(5x - 10)(x + 8)(x - 7) > 0
5x - 10 = 0
x₁ = 2
x + 8 = 0
x₂ = - 8
x - 7 = 0
x₃ = 7
         -                    +                      -                        +
-------------------------------------------------------------------------------------------->
                - 8                     2                         7                       x
x∈(-8;2)  (7;+∞)

Answer 2

To solve the inequality $(5x-10)/((x+8)(x-7)) > 0$, we can use a sign chart or interval notation.

Let's break it down into three parts by considering the sign of each factor in the denominator: $x+8$, $x-7$, and $(5x-10)$.

1. $x+8 > 0$: Solving this inequality, we find $x > -8$. This means that $x$ is positive to the right of -8.

2. $x-7 > 0$: Solving this inequality, we find $x > 7$. This means that $x$ is positive to the right of 7.

3. $5x-10 > 0$: Solving this inequality, we find $x > 2$. This means that $x$ is positive to the right of 2.

Now, let's consider all the cases:

Case 1: $x < -8$: In this case, all three factors are negative. Dividing a negative number by negative factors will result in a positive value. Therefore, the inequality $(5x-10)/((x+8)(x-7)) > 0$ is true when $x < -8$.

Case 2: $-8 < x < 2$: In this case, $x+8$ is positive, $x-7$ is negative, and $5x-10$ is negative. Dividing a negative number by positive factors will result in a negative value. Therefore, the inequality $(5x-10)/((x+8)(x-7)) > 0$ is false when $-8 < x < 2$.

Case 3: $x > 2$: In this case, all three factors are positive. Dividing a positive number by positive factors will result in a positive value. Therefore, the inequality $(5x-10)/((x+8)(x-7)) > 0$ is true when $x > 2$.

To summarize: $(5x-10)/((x+8)(x-7)) > 0$ is true when $x < -8$ or $x > 2$.

In interval notation, the solution is $(-∞, -8) ∪ (2, ∞)$.