Question

(2 - 8x)(x + 1) >( 1 - 4x)(x + 2)​

Answer 1

Ответ:

Пошаговое объяснение:

(2 - 8x)(x + 1) >( 1 - 4x)(x + 2)​

-8x²-6x+2>-4x²-7x+2

-8x²-6x+2+4x²+7x-2>0

-4x²+x>0

4x²-x<0

нули:

x=0

x=1/4

Получим:

x∈ ( 0 ; 1/4 )

Answer 2

To solve the inequality $(2 - 8x)(x + 1) > (1 - 4x)(x + 2)$, you can follow these steps:

1. First, expand both sides of the inequality:

$(2 - 8x)(x + 1) > (1 - 4x)(x + 2)$

Expanding both sides:

$2x + 2 - 8x^2 - 8x > x + 2 - 4x^2 - 8x$

2. Rearrange the terms and simplify:

$2x + 2 - 8x^2 - 8x > x + 2 - 4x^2 - 8x$

Combine like terms on both sides:

$2x - 8x - 8x^2 + 2 > x - 4x^2 - 8x + 2$

3. Move all terms to one side to set the inequality to zero:

$2x - 8x - 8x^2 + 2 - (x - 4x^2 - 8x + 2) > 0$

Now, simplify:

$2x - 8x - 8x^2 + 2 - x + 4x^2 + 8x - 2 > 0$

4. Combine like terms:

$-8x^2 - x + 2 - 2 > 0$

5. Further simplify:

$-8x^2 - x > 0$

6. Factor out a negative sign to make it easier to work with:

$-1(8x^2 + x) > 0$

7. Divide both sides by -1. Remember that when you divide or multiply an inequality by a negative number, you must reverse the direction of the inequality:

$8x^2 + x < 0$

8. Now, you need to find the values of $x$ that satisfy this inequality. To do that, you can find the critical points by setting the expression equal to zero and solving for $x$:

$8x^2 + x = 0$

Factor:

$x(8x + 1) = 0$

Now, you have two possible critical points:

a) $x = 0$ b) $8x + 1 = 0$

For $8x + 1 = 0$, solving for $x$:

$8x = -1$ $x = -1/8$

So, the critical points are $x = 0$ and $x = -1/8$.

9. Plot these critical points on the number line and test points in each interval to determine the solution to the inequality:

• Test a point in the interval $(-∞, -1/8)$, let's say $x = -1$: $8(-1)^2 + (-1) = 9$, which is positive.

• Test a point in the interval $(-1/8, 0)$, let's say $x = -1/4$: $8(-1/4)^2 + (-1/4) = 1/4 - 1/4 = 0$, which is zero.

• Test a point in the interval $(0, ∞)$, let's say $x = 1$: $8(1)^2 + (1) = 8 + 1 = 9$, which is positive.

Since the inequality $8x^2 + x < 0$ is negative between $x = -1/8$ and $x = 0$, the solution to the original inequality $(2 - 8x)(x + 1) > (1 - 4x)(x + 2)$ is:

$-1/8 < x < 0$