Question

(x – 2)(x + 3) / (1 – x) ≥ 0

Answer 1

Ответ:

х$\neq$1

х=2, х=-3

 +       -         +        -

__._____о___._________> x

  -3          1      2

интервалы там где "+"

(-∞;-3]U(1;2]

Answer 2

Ответ: (-∞; -3] и (1; 2]

В решение использован метод интервалов.

Answer 3

To solve the inequality $\frac{(x - 2)(x + 3)}{1 - x} \geq 0$, we can follow these steps:

1. Determine the critical points by finding where the numerator and denominator equal zero:

   • Numerator: $(x - 2)(x + 3) = 0$ Set each factor equal to zero:

     • $x - 2 = 0$ implies $x = 2$
     • $x + 3 = 0$ implies $x = -3$

   • Denominator: $1 - x = 0$ Solve for $x$:

     • $1 - x = 0$ implies $x = 1$

2. Use these critical points to create intervals on the number line. These intervals are determined by the critical points and the sign of the expression in each interval:

   • Interval 1: $(-∞, -3)$

     • Choose a test point $x < -3$, for example, $x = -4$.
     • Plug it into the expression: $\frac{(x - 2)(x + 3)}{1 - x} = \frac{(-4 - 2)(-4 + 3)}{1 + 4} = \frac{-6 \cdot -1}{5} = \frac{6}{5}$
     • Since $\frac{6}{5}$ is positive, this interval is positive.

   • Interval 2: $(-3, 1)$

     • Choose a test point $x$ between -3 and 1, for example, $x = 0$.
     • Plug it into the expression: $\frac{(x - 2)(x + 3)}{1 - x} = \frac{(0 - 2)(0 + 3)}{1 - 0} = \frac{-6}{1} = -6$
     • Since -6 is negative, this interval is negative.

   • Interval 3: $(1, 2)$

     • Choose a test point $x$ between 1 and 2, for example, $x = 1.5$.
     • Plug it into the expression: $\frac{(x - 2)(x + 3)}{1 - x} = \frac{(1.5 - 2)(1.5 + 3)}{1 - 1.5} = \frac{-0.5 \cdot 4.5}{-0.5} = 4.5$
     • Since 4.5 is positive, this interval is positive.

   • Interval 4: $(2, ∞)$

     • Choose a test point $x > 2$, for example, $x = 3$.
     • Plug it into the expression: $\frac{(x - 2)(x + 3)}{1 - x} = \frac{(3 - 2)(3 + 3)}{1 - 3} = \frac{1 \cdot 6}{-2} = -3$
     • Since -3 is negative, this interval is negative.

3. Now, combine the information from the intervals:

   • The expression is positive in intervals 1 and 3.
   • The expression is negative in intervals 2 and 4.

4. To satisfy the inequality $\frac{(x - 2)(x + 3)}{1 - x} \geq 0$, we need the expression to be greater than or equal to zero. This means that we are looking for the intervals where the expression is non-negative (positive or zero).

So, the solution to the inequality is: $x \in (-\infty, -3] \cup [1, 2]$

These are the values of $x$ that make the expression $\frac{(x - 2)(x + 3)}{1 - x}$ greater than or equal to zero.