Question

(2,5-x)(2x+3)(x+4)>0 решите уравнение

Answer 1

Ответ:

это и есть ответ, он на картинке

Answer 2

To solve the inequality $(2,5-x)(2x+3)(x+4) > 0$, we need to find the intervals of $x$ that satisfy the inequality. To do that, we can use the concept of interval notation and test the sign of the expression in each interval.

Step 1: Find the critical points by setting each factor equal to zero and solving for $x$:

1. $2.5 - x = 0 \Rightarrow x = 2.5$
2. $2x + 3 = 0 \Rightarrow x = -1.5$
3. $x + 4 = 0 \Rightarrow x = -4$

Step 2: Plot these critical points on a number line and choose test points in the regions created by these critical points. We'll test the sign of the expression $(2.5 - x)(2x + 3)(x + 4)$ for each interval:

• Test point $x = -5$ (a value less than -4)
• Test point $x = -2$ (a value between -4 and -1.5)
• Test point $x = 0$ (a value between -1.5 and 2.5)
• Test point $x = 3$ (a value greater than 2.5)

Step 3: Determine the sign of the expression for each test point:

1. $x = -5$: $(2.5 - (-5))(2(-5) + 3)(-5 + 4) = (7.5)(-7)(-1) = 52.5$, which is positive.
2. $x = -2$: $(2.5 - (-2))(2(-2) + 3)(-2 + 4) = (4.5)(-1)(2) = -9$, which is negative.
3. $x = 0$: $(2.5 - 0)(2(0) + 3)(0 + 4) = (2.5)(3)(4) = 30$, which is positive.
4. $x = 3$: $(2.5 - 3)(2(3) + 3)(3 + 4) = (-0.5)(9)(7) = -31.5$, which is negative.

Step 4: Analyze the intervals:

• The expression is positive in the interval $-\infty < x < -4$ and $0 < x < 2.5$.
• The expression is negative in the interval $-4 < x < -1.5$ and $2.5 < x < \infty$.

Step 5: Write the solution in interval notation:

The solution to the inequality $(2.5 - x)(2x + 3)(x + 4) > 0$ is:

$x \in (-\infty, -4) \cup (0, 2.5)$

So, any value of $x$ that falls within the intervals $(- \infty, -4)$ and $(0, 2.5)$ will satisfy the inequality.