Certainly! Let's analyze and solve the inequality [tex]\((x+1)(2-x) < 0\)[/tex] step-by-step.
### Step 1: Identify the expressions inside the products and determine the sign changes
We have the product [tex]\((x+1)(2-x)\)[/tex]. To solve the inequality, we need to determine the x-values for which this product is negative. We'll start by identifying the roots of each factor:
1. [tex]\(x + 1 = 0\)[/tex]
[tex]\[
x = -1
\][/tex]
2. [tex]\(2 - x = 0\)[/tex]
[tex]\[
x = 2
\][/tex]
These roots, [tex]\(x = -1\)[/tex] and [tex]\(x = 2\)[/tex], are critical points because they divide the number line into sections where the sign of the product [tex]\((x+1)(2-x)\)[/tex] could change.
### Step 2: Test intervals between the roots
We will analyze the sign of [tex]\((x+1)(2-x)\)[/tex] in the intervals defined by these roots: [tex]\( (-\infty, -1) \)[/tex], [tex]\( (-1, 2) \)[/tex], and [tex]\( (2, \infty) \)[/tex].
1. Interval [tex]\((- \infty, -1)\)[/tex]:
Pick a point [tex]\(x = -2\)[/tex]:
[tex]\[
(x + 1)(2 - x) = (-2 + 1)(2 - (-2)) = (-1)(4) = -4
\][/tex]
The product is negative in this interval.
2. Interval [tex]\((-1, 2)\)[/tex]:
Pick a point [tex]\(x = 0\)[/tex]:
[tex]\[
(x + 1)(2 - x) = (0 + 1)(2 - 0) = (1)(2) = 2
\][/tex]
The product is positive in this interval.
3. Interval [tex]\((2, \infty)\)[/tex]:
Pick a point [tex]\(x = 3\)[/tex]:
[tex]\[
(x + 1)(2 - x) = (3 + 1)(2 - 3) = (4)(-1) = -4
\][/tex]
The product is negative in this interval.
### Step 3: Combine the intervals where the product is negative
The inequality [tex]\((x+1)(2-x) < 0\)[/tex] holds true in the intervals where the product is negative. From the testing above:
- The product is negative when [tex]\(x \in (-\infty, -1)\)[/tex]
- The product is negative when [tex]\(x \in (2, \infty)\)[/tex]
### Step 4: Write the solution in interval notation
Combining these intervals, we get the solution to the inequality:
[tex]\[
x \in (-\infty, -1) \cup (2, \infty)
\][/tex]
### Conclusion
The solution to the inequality [tex]\((x+1)(2-x) < 0\)[/tex] is:
[tex]\[
(-\infty < x < -1) \cup (2 < x < \infty)
\][/tex]
Which in mathematical notation is:
[tex]\[
x \in (-\infty, -1) \cup (2, \infty)
\][/tex]
This completes the solution to the inequality.
