Solve for inequality using the interval method: (2x+1)(3-x) < 0

We are asked to solve the inequality \((2x + 1)(3 - x) < 0\) using the interval method. Let's go through the steps to find the solution.

### Step 1: Find the critical points

To begin, solve for when each factor equals zero:
- \(2x + 1 = 0\)
  \[
  2x = -1 \quad \Rightarrow \quad x = -\frac{1}{2}
  \]
- \(3 - x = 0\)
  \[
  x = 3
  \]

These values split the real line into intervals. The critical points are \(x = -\frac{1}{2}\) and \(x = 3\).

### Step 2: Determine intervals

The critical points divide the real line into three intervals:
- \( (-\infty, -\frac{1}{2}) \)
- \( (-\frac{1}{2}, 3) \)
- \( (3, \infty) \)

### Step 3: Test the sign of each interval

Now, we test the sign of the expression \((2x + 1)(3 - x)\) in each interval.

1. **Interval \( (-\infty, -\frac{1}{2}) \)**:
   - Pick a test point, say \(x = -1\).
   - \((2(-1) + 1)(3 - (-1)) = (-2 + 1)(3 + 1) = (-1)(4) = -4\)
   - The product is negative in this interval.

2. **Interval \( (-\frac{1}{2}, 3) \)**:
   - Pick a test point, say \(x = 0\).
   - \((2(0) + 1)(3 - 0) = (1)(3) = 3\)
   - The product is positive in this interval.

3. **Interval \( (3, \infty) \)**:
   - Pick a test point, say \(x = 4\).
   - \((2(4) + 1)(3 - 4) = (8 + 1)(-1) = 9(-1) = -9\)
   - The product is negative in this interval.

### Step 4: Analyze the inequality

We are looking for where \((2x + 1)(3 - x) < 0\), i.e., where the product is negative. From the sign analysis:
- The expression is negative on the intervals \( (-\infty, -\frac{1}{2}) \) and \( (3, \infty) \).

### Step 5: Final solution

Therefore, the solution to the inequality \((2x + 1)(3 - x) < 0\) is:

\[
x \in (-\infty, -\frac{1}{2}) \cup (3, \infty)
\]

Let me know if you'd like further details on any part of the solution.

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Here are 5 related questions:
1. How do you solve inequalities involving quadratic expressions?
2. What changes in the solution if the inequality sign is \(\leq\) instead of \(<\)?
3. How do we interpret solutions when the inequality is \(> 0\) instead of \(< 0\)?
4. How does interval notation work, and what do the symbols \(( )\) and \([ ]\) represent?
5. How do you approach solving inequalities when there are more than two factors involved?

**Tip:** Always verify your intervals by choosing test points within them to check the sign of the expression.

Learn how to solve the inequality (2x+1)(3-x) < 0 using the interval method with a step-by-step solution. This problem demonstrates key algebraic concepts such as critical points, interval testing, and sign analysis. Suitable for students in Grades 9-11.

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