Question

(x - 3)(4 - 2x) >= 0

Answer 1

Ответ:

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Answer 2

To solve the inequality $(x - 3)(4 - 2x) \geq 0$, we'll follow these steps:

1. Find the critical points by setting each factor equal to zero and solving for $x$.
2. Create intervals using these critical points.
3. Test points within each interval to determine the sign of the expression.
4. Determine the intervals that satisfy the inequality.

Let's go through it step by step:

1. Find the critical points:

   $(x - 3) = 0$ gives $x = 3$.

   $(4 - 2x) = 0$ gives $x = 2$.

2. Create intervals:

   These critical points divide the number line into three intervals: $(-∞, 2)$, $(2, 3)$, and $(3, ∞)$.

3. Test points within each interval:

   • For the interval $(-∞, 2)$, we can choose a test point, say $x = 0$. Plug it into the original inequality: $(0 - 3)(4 - 2(0)) \geq 0$ which simplifies to $-12 \geq 0$. This is false.

   • For the interval $(2, 3)$, we can choose a test point, say $x = 2.5$. Plug it into the original inequality: $(2.5 - 3)(4 - 2(2.5)) \geq 0$ which simplifies to $1 \geq 0$. This is true.

   • For the interval $(3, ∞)$, we can choose a test point, say $x = 4$. Plug it into the original inequality: $(4 - 3)(4 - 2(4)) \geq 0$ which simplifies to $-4 \geq 0$. This is false.

4. Determine the intervals that satisfy the inequality:

   The solution to the inequality is $x$ belonging to the interval $(2, 3)$, or in interval notation: $x \in (2, 3)$.

Please note that the inequality is not strict (greater than), so the endpoints $x = 2$ and $x = 3$ are not included in the solution set.