Question

X(3x - 6)*(1- x) ≥ 0

Answer 1

Ответ приложен в фотографии

Answer 2

To solve the inequality $X(3x - 6)(1 - x) \geq 0$, you need to determine the intervals of $x$ values for which the inequality is satisfied. The inequality involves a quadratic expression and factors, so we can approach it using the concept of intervals and signs.

Step 1: Find the critical points by setting each factor equal to zero and solving for $x$:

1. $X = 0$
2. $3x - 6 = 0 \Rightarrow x = 2$
3. $1 - x = 0 \Rightarrow x = 1$

Step 2: Create a sign table to analyze the intervals and the sign of the expression $X(3x - 6)(1 - x)$:

Interval	$X$	$3x - 6$	$1 - x$	$X(3x - 6)(1 - x)$
$x < 0$	$-$	$-$	$-$	$-$
$0 < x < 1$	$-$	$-$	$+$	$+$
$1 < x < 2$	$-$	$+$	$+$	$-$
$x > 2$	$+$	$+$	$-$	$-$

Step 3: Interpret the sign table and find the intervals where $X(3x - 6)(1 - x) \geq 0$:

The inequality is satisfied in the intervals:

1. $0 < x < 1$
2. $x > 2$

So, the solution to the inequality is $0 < x < 1$ and $x > 2$. In interval notation, the solution is $x \in (0, 1) \cup (2, \infty)$.