Question

2x+8/x-7<0 решите неравенства

Answer 1

Запишем все возможные исходы когда частное < 0:
2x+8<0 и x-7>0 или 2x+8>0 и x-7<0⇒x<-4 и x>7 или x>-4 и x<7⇒X∈∅ и X∈(-4;7)⇒Окончательный ответ - X∈(-4;7)

Answer 2

To solve the inequality $2x + \frac{8}{x} - 7 < 0$, follow these steps:

Step 1: Find the critical points by setting the expression inside the inequality to zero and solving for $x$: $2x + \frac{8}{x} - 7 = 0$.

Step 2: Solve for $x$ to find the critical points: $2x^2 - 7x + 8 = 0$.

To solve this quadratic equation, you can use factoring or the quadratic formula:

Factoring: $2x^2 - 7x + 8 = (2x - 1)(x - 8) = 0$.

So the critical points are $x = \frac{1}{2}$ and $x = 8$.

Step 3: Create a number line and plot the critical points on it:

cssCopy code
-----o---------o-----
   1/2         8

Step 4: Test a point from each interval to determine the sign of the expression $2x + \frac{8}{x} - 7$ within that interval.

Test $x = 0$: $2(0) + \frac{8}{0} - 7$ is undefined because division by zero is not allowed.

Test $x = 1$: $2(1) + \frac{8}{1} - 7 = 3 > 0$.

Test $x = 5$: $2(5) + \frac{8}{5} - 7 = \frac{6}{5} > 0$.

Step 5: Determine the sign of the expression $2x + \frac{8}{x} - 7$ in each interval and write the solution:

• In the interval $x < \frac{1}{2}$, the expression is negative.
• In the interval $\frac{1}{2} < x < 8$, the expression is positive.
• In the interval $x > 8$, the expression is positive.

Step 6: Write the final solution for the inequality: $2x + \frac{8}{x} - 7 < 0 \quad \text{when} \quad x < \frac{1}{2} \quad \text{or} \quad 8 < x.$

So, the solution to the inequality is $x < \frac{1}{2}$ or $x > 8$.