Question

B) (3x - 2)(x - 4)(3 - 2x) < 0;​

Answer 1

Сверьте свой ответ с моим.

х=2/3, х=4; х=1.5

______2/3_____1.5_______4_____

 +                  -                +              -

х∈(2/3;1.5)∪(4;+∞)

Answer 2

To solve the inequality (3x - 2)(x - 4)(3 - 2x) < 0, we can use a method called the sign chart or the number line method.

First, let's find the critical points of the expression, which are the values of x where the expression is equal to zero.

Setting each factor to zero individually:

3x - 2 = 0 x = 2/3

x - 4 = 0 x = 4

3 - 2x = 0 -2x = -3 x = 3/2

So the critical points are x = 2/3, x = 4, and x = 3/2.

Now, we can create a sign chart or use a number line to determine the intervals where the expression is positive or negative.

On a number line, we mark the critical points:

2/3 3/2 4 | | |

Next, we choose test points from each interval and substitute them into the expression to determine the sign.

For example, let's choose x = 0 as a test point in the interval (-∞, 2/3):

(3(0) - 2)(0 - 4)(3 - 2(0)) = (-2)(-4)(3) = 24

Since the result is positive, we put a "+" sign above the interval (-∞, 2/3).

Next, let's choose x = 1 as a test point in the interval (2/3, 3/2):

(3(1) - 2)(1 - 4)(3 - 2(1)) = (1)(-3)(1) = -3

Since the result is negative, we put a "-" sign above the interval (2/3, 3/2).

Finally, let's choose x = 5 as a test point in the interval (4, ∞):

(3(5) - 2)(5 - 4)(3 - 2(5)) = (13)(1)(-7) = -91

Since the result is negative, we put a "-" sign above the interval (4, ∞).

Now, we can see the sign pattern:

2/3 3/2 4 | + - - |

The inequality (3x - 2)(x - 4)(3 - 2x) < 0 is true when the expression is negative, which occurs in the intervals (2/3, 3/2) and (4, ∞).

Therefore, the solution to the inequality is:

2/3 < x < 3/2 or x > 4