Question

X²+10x/10-2x+5/2<=20!!!!!!!!!​

Answer 1

Ответ:

$\frac{x^{2} +10x}{10-2x+5} :2\leq 20$

$\frac{x^{2} +10x}{10-2x+5} :\frac{2}{1} -\frac{20}{1} \leq 0$

$\frac{x^{2} +10x}{10-2x+5} *\frac{1}{2} -\frac{20}{1} \leq 0$

$\frac{x^{2} +10x}{20-4x+10} -\frac{20}{1} \leq 0$

$\frac{x^{2} +10x}{20-4x+10} -\frac{20*(20-4x+10)}{1*(20-4x+10)} \leq 0$

$\frac{x^{2} +10x}{20-4x+10} -\frac{400-80x+200}{20-4x+10} \leq 0$

$\frac{x^{2} +10x-400+80x-200}{20-4x+10} \leq 0$

$\frac{x^{2} +90x-600}{30-4x} \leq 0$

Приравняем числитель и знаменатель к нулю:

1). 30-4x=0

4x=30

x=7,5

2). x²+90x-600=0

D = 90²-4·(-600) = 8100+2400 = 10500

$x_{1} =\frac{ -90+\sqrt{1,05*100*100} }{2} = \frac{-90+100+\sqrt{1,05} }{2} = \frac{10+\sqrt{1,05} }{2}$

$x_{2} =\frac{ -90-\sqrt{1,05*100*100} }{2} = \frac{-90-100-\sqrt{1,05} }{2} = \frac{-190-\sqrt{1,05} }{2}$

Запишем ответ.

x из знаменателя пишем с круглой скобкой.

x∈ $[\frac{-190-\sqrt{x} 1,05}{2} ; \frac{10+\sqrt{1,05} }{2} ]$ ∪ (7,5;+∞)

Answer 2

To solve the inequality x² + 10x/10 - 2x + 5/2 <= 20, you can follow these steps:

1. First, simplify the expression: x² + x - 2x + 2.5 <= 20

2. Combine like terms: x² - x + 2.5 <= 20

3. Subtract 20 from both sides of the inequality to isolate the x² term: x² - x + 2.5 - 20 <= 0

4. Combine the constants on the left side: x² - x - 17.5 <= 0

5. Now, we need to solve this quadratic inequality. To do that, you can find the critical points by setting the expression equal to zero: x² - x - 17.5 = 0

6. Factor the quadratic: (x - 4.5)(x + 3.5) = 0

7. Find the solutions to the equation: x = 4.5 and x = -3.5

These are the critical points where the expression equals zero. Now, you can create a sign chart to test intervals around these points.

• Choose test values within each interval: Let's test values in the intervals (-∞, -3.5), (-3.5, 4.5), and (4.5, ∞).
• For the interval (-∞, -3.5), choose x = -4.
• For the interval (-3.5, 4.5), choose x = 0.
• For the interval (4.5, ∞), choose x = 5.

8. Plug in these test values into the inequality and check the signs:

• For x = -4: (-4)² - (-4) - 17.5 is positive.

• For x = 0: (0)² - (0) - 17.5 is negative.

• For x = 5: (5)² - 5 - 17.5 is positive.

Now, analyze the sign chart:

• In the interval (-∞, -3.5), the expression is positive.
• In the interval (-3.5, 4.5), the expression is negative.
• In the interval (4.5, ∞), the expression is positive.

Since you want the inequality to be less than or equal to 0, you're looking for the intervals where the expression is non-positive. Therefore, the solution to the inequality is:

x in (-3.5, 4.5]

So, the values of x that satisfy the inequality are all real numbers between -3.5 (exclusive) and 4.5 (inclusive).