Question

Решить неравенство x^2-5x-6<0

Answer 1

X1x2=-6
x1+x2=5

Значит x1=6
x2=-1

Answer 2

Ответ: x принадлежит промежутку (-1; 6)

Answer 3

To solve the inequality $x^2 - 5x - 6 < 0$, we can use a method called "sign chart" or "interval notation." First, we need to find the critical points where the expression equals zero, and then we can analyze the sign of the expression within each interval created by these critical points.

Step 1: Find the critical points (where $x^2 - 5x - 6 = 0$): To find the critical points, we set the expression equal to zero and solve for $x$: $x^2 - 5x - 6 = 0.$

This is a quadratic equation that can be factored: $(x - 6)(x + 1) = 0.$

So the critical points are $x = 6$ and $x = -1$.

Step 2: Create a sign chart:

Now, we create a sign chart to determine the sign of the expression $x^2 - 5x - 6$ in different intervals.

Interval	$x^2 - 5x - 6$
$x < -1$	?
$-1 < x < 6$	?
$x > 6$	?

Step 3: Test a point in each interval:

We can now pick a test point within each interval to determine the sign of the expression.

1. Let's test $x = -2$ (a number less than -1): $(\text{-2})^2 - 5(\text{-2}) - 6 = 4 + 10 - 6 = 8 > 0.$

2. Let's test $x = 0$ (a number between -1 and 6): $0^2 - 5 \times 0 - 6 = -6 < 0.$

3. Let's test $x = 7$ (a number greater than 6): $7^2 - 5 \times 7 - 6 = 49 - 35 - 6 = 8 > 0.$

Step 4: Analyze the sign chart:

Based on the test results, we can fill in the sign chart:

Interval	$x^2 - 5x - 6$
$x < -1$	$> 0$
$-1 < x < 6$	$< 0$
$x > 6$	$> 0$

Step 5: Determine the solution:

The inequality is satisfied when $x^2 - 5x - 6 < 0$, which means we are looking for the interval where the expression is negative. From the sign chart, we see that the expression is negative when $-1 < x < 6$.

So, the solution to the inequality $x^2 - 5x - 6 < 0$ is $x$ belongs to the open interval $(-1, 6)$. In interval notation, the solution is $x \in (-1, 6)$.