Question

1) x^2+5x<0 2) x^2<25 3) 2x^2-3x-5<0

Answer 1

Ответ:

1. x^2 +5x<0

x * (x+5)<0

{x<0

{x+5>0

{x>0

{x+5<0

{x>0

{x<-5

x э (-5;0)

2. x^2<25

/x/<5

x<5, x>=0

x э [0,5)

x>-5, x<0

x э [0,5)

x э (-5,0)

3. 2x^2-3x-5<0

2x^2+2x-5x-5<0

2x×(x+1)-5(x+1)<0

(x+1)×(2x-5)<0

{x+1<0

{2x-5>0

{x+1>0

{2x-5<0

{x<-1

{x> 5/2

{x>-1

{x<5/2

Answer 2

To solve these inequalities, let's break them down one by one:

1. $x^2 + 5x < 0$

Let's factor the left side:

$x(x + 5) < 0$

Now we have two factors, $x$ and $x + 5$. We need to consider the sign of each factor.

$x$ is zero when $x = 0$ and $x + 5$ is zero when $x = -5$.

Now, we can create a sign chart:

markdownCopy code
     |  x  | x + 5 | x(x + 5)
---------------------------
x < -5 |  -  |   -   |    +
-5 < x < 0 |  -  |   +   |    -
x > 0  |  +  |   +   |    +

The solution to $x^2 + 5x < 0$ is $x < -5$ or $0 < x < 5$.

2. $x^2 < 25$

Let's factor the right side:

$x^2 - 25 < 0$

$(x - 5)(x + 5) < 0$

We have two factors, $x - 5$ and $x + 5$. Again, consider the sign of each factor.

$x - 5$ is zero when $x = 5$ and $x + 5$ is zero when $x = -5$.

Create a sign chart:

markdownCopy code
     |  x - 5  | x + 5 | (x - 5)(x + 5)
---------------------------
x < -5 |   -    |   -   |       +
-5 < x < 5 |   -    |   +   |       -
x > 5  |   +    |   +   |       +

The solution to $x^2 < 25$ is $-5 < x < 5$.

3. $2x^2 - 3x - 5 < 0$

This quadratic inequality is a bit more complicated. Let's find the roots first:

$2x^2 - 3x - 5 = 0$

This quadratic equation can be factored as $(2x + 1)(x - 5) = 0$, so the roots are $x = -\frac{1}{2}$ and $x = 5$.

Create a sign chart:

markdownCopy code
     |  2x + 1 |  x - 5 |  2x^2 - 3x - 5
-----------------------------------
x < -1/2 |   -    |   -   |         +
-1/2 < x < 5 |   +    |   -   |         -
x > 5  |   +    |   +   |         +

The solution to $2x^2 - 3x - 5 < 0$ is $-\frac{1}{2} < x < 5$.

If you have any further questions, please, let me know.