Question

(x - 1)(x - 4) > 0 (3x + 2)(4x + 3) >= 0 (x - 2)(4 - x) > 0 . (3x + 5)(3 - 4x) <= 0 .

Answer 1

Ответ:

1. 9

2. 13

3. 7

4. 16

Мне лень решать вроде так

Answer 2

To solve these inequalities, we can follow these steps:

1. Find the critical points by setting each factor equal to zero and solving for x.
2. Create sign charts to determine the intervals where each factor is positive or negative.
3. Combine the information from all factors to determine the intervals where the original inequality is satisfied.

Let's solve each inequality step by step:

1. $(x - 1)(x - 4) > 0$

First, find the critical points: $x - 1 = 0$ => $x = 1$ $x - 4 = 0$ => $x = 4$

Now create a sign chart for $(x - 1)(x - 4)$:

Interval	(x - 1)(x - 4)
$x < 1$	(-)(-) = (+)
$1 < x < 4$	(+)(-) = (-)
$x > 4$	(+)(+) = (+)

So, the solution to $(x - 1)(x - 4) > 0$ is $x < 1$ or $x > 4$.

2. $(3x + 2)(4x + 3) >= 0$

Find the critical points: $3x + 2 = 0$ => $x = -2/3$ $4x + 3 = 0$ => $x = -3/4$

Create a sign chart for $(3x + 2)(4x + 3)$:

Interval	(3x + 2)(4x + 3)
$x < -3/4$	(-)(-) = (+)
$-3/4 < x < -2/3$	(-)(+) = (-)
$x > -2/3$	(+)(+) = (+)

So, the solution to $(3x + 2)(4x + 3) >= 0$ is $x < -3/4$ or $x > -2/3$.

3. $(x - 2)(4 - x) > 0$

Find the critical points: $x - 2 = 0$ => $x = 2$ $4 - x = 0$ => $x = 4$

Create a sign chart for $(x - 2)(4 - x)$:

Interval	(x - 2)(4 - x)
$x < 2$	(-)(+) = (-)
$2 < x < 4$	(+)(+) = (+)
$x > 4$	(+)(-) = (-)

So, the solution to $(x - 2)(4 - x) > 0$ is $2 < x < 4$.

4. $(3x + 5)(3 - 4x) <= 0$

Find the critical points: $3x + 5 = 0$ => $x = -5/3$ $3 - 4x = 0$ => $x = 3/4$

Create a sign chart for $(3x + 5)(3 - 4x)$:

Interval	(3x + 5)(3 - 4x)
$x < -5/3$	(-)(+) = (-)
$-5/3 < x < 3/4$	(-)(-) = (+)
$x > 3/4$	(+)(-) = (-)

So, the solution to $(3x + 5)(3 - 4x) <= 0$ is $-5/3 < x < 3/4$.

Now, to find the combined solution for all inequalities, you can look for the overlapping intervals:

• For the first inequality, $x < 1$ or $x > 4$.
• For the second inequality, $x < -3/4$ or $x > -2/3$.
• For the third inequality, $2 < x < 4$.
• For the fourth inequality, $-5/3 < x < 3/4$.

To find the combined solution, you need to consider where all conditions overlap. The overlapping intervals are $2 < x < 3/4$ and $x > 4$. So, the solution to the system of inequalities is $x > 4$.