(x−3)(2x+7)>0.

To solve the inequality \((x - 3)(2x + 7) > 0\), we need to find the values of \(x\) for which this product is positive.

Step 1: Determine the critical points.

Set each factor equal to zero:

1. \(x - 3 = 0 \Rightarrow x = 3\)
2. \(2x + 7 = 0 \Rightarrow 2x = -7 \Rightarrow x = -\frac{7}{2}\)

The critical points are \(x = 3\) and \(x = -\frac{7}{2}\).

Step 2: Test the intervals determined by these critical points.

The critical points divide the number line into three intervals:

• \( (-\infty, -\frac{7}{2}) \)
• \( (-\frac{7}{2}, 3) \)
• \( (3, \infty) \)

Step 3: Test a point from each interval.

1. For the interval \( (-\infty, -\frac{7}{2}) \), let’s choose \(x = -4\): \[ (-4 - 3)(2(-4) + 7) = (-7)(-8 + 7) = (-7)(-1) = 7 > 0 \] (The product is positive.)

2. For the interval \( (-\frac{7}{2}, 3) \), let’s choose \(x = 0\): \[ (0 - 3)(2(0) + 7) = (-3)(7) = -21 < 0 \] (The product is negative.)

3. For the interval \( (3, \infty) \), let’s choose \(x = 4\): \[ (4 - 3)(2(4) + 7) = (1)(8 + 7) = (1)(15) = 15 > 0 \] (The product is positive.)

Step 4: Combine the results.

The product \((x - 3)(2x + 7)\) is positive in the intervals:

• \( (-\infty, -\frac{7}{2}) \)
• \( (3, \infty) \)

Step 5: Write the solution.

The solution to the inequality \((x - 3)(2x + 7) > 0\) is: \[ x \in (-\infty, -\frac{7}{2}) \cup (3, \infty) \]