Question

(х+8)(х-4)<0 полное решение

Answer 1

.......,.................

Answer 2

To find the solution to the inequality $(x+8)(x-4) < 0$, we need to determine the intervals of values for $x$ that satisfy the inequality. To do this, we can use a method called "sign chart" or "interval notation."

Step 1: Find the critical points The critical points are the values of $x$ that make the expression $x+8$ or $x-4$ equal to zero. So, set each factor to zero and solve for $x$:

$x + 8 = 0 \Rightarrow x = -8$

$x - 4 = 0 \Rightarrow x = 4$

Step 2: Create a sign chart

Now, we'll create a sign chart with the critical points and test intervals:

$$\begin{array}{|c|c|c|c|c|} \hline \text{Interval} & ( -\infty, -8) & (-8, 4) & (4, \infty) \\ \hline x + 8 & - & + & + \\ \hline x - 4 & - & - & + \\ \hline (x + 8)(x - 4) & + & - & + \\ \hline \end{array}$$

Step 3: Interpret the sign chart

The sign chart shows the sign of each factor and the sign of the entire expression $(x + 8)(x - 4)$ within each interval. The expression is positive in the intervals where both factors have the same sign (either both positive or both negative) and negative when the factors have opposite signs.

Step 4: Determine the solution

The inequality $(x + 8)(x - 4) < 0$ is satisfied when the expression $(x + 8)(x - 4)$ is negative. From the sign chart, we see that the expression is negative in the interval $(-8, 4)$.

Thus, the solution to the inequality is $x \in (-8, 4)$. In interval notation, the solution is $x \in (-8, 4)$. This means any value of $x$ within the open interval $(-8, 4)$ will satisfy the inequality.