Question

Solve the quadratic inequality x2−7x+12>0 by factoring.(1 point) In interval notation, the solutions to the quadratic inequality are (−∞ ,) ∪ (,∞ ).

Answer 1

To solve the quadratic inequality \(x^2 - 7x + 12 > 0\) by factoring, we first need to factor the quadratic expression.

1. **Factor the quadratic**:
The expression \(x^2 - 7x + 12\) can be factored as follows:
\[
x^2 - 7x + 12 = (x - 3)(x - 4)
\]

2. **Set the factors greater than zero**:
We need to find where \((x - 3)(x - 4) > 0\).

3. **Determine the critical points**:
The critical points occur when \( (x - 3) = 0\) and \( (x - 4) = 0\). This gives us:
\[
x = 3 \quad \text{and} \quad x = 4
\]

4. **Test the intervals**:
We will test the sign of \((x - 3)(x - 4)\) in the intervals determined by the critical points \(3\) and \(4\):
- **Interval 1**: \( (-\infty, 3) \)
- **Interval 2**: \( (3, 4) \)
- **Interval 3**: \( (4, \infty) \)

- **Test \(x = 2\)** (in \( (-\infty, 3) \)):
\[
(2 - 3)(2 - 4) = (-1)(-2) = 2 > 0
\]

- **Test \(x = 3.5\)** (in \( (3, 4) \)):
\[
(3.5 - 3)(3.5 - 4) = (0.5)(-0.5) = -0.25 < 0
\]

- **Test \(x = 5\)** (in \( (4, \infty) \)):
\[
(5 - 3)(5 - 4) = (2)(1) = 2 > 0
\]

5. **Conclusion from the test**:
We find that \((x - 3)(x - 4) > 0\) in the intervals \((-∞, 3)\) and \((4, ∞)\).

6. **Write the solution in interval notation**:
Therefore, the solution to the quadratic inequality \(x^2 - 7x + 12 > 0\) is:
\[
(-\infty, 3) \cup (4, \infty)
\]

Thus, in interval notation, the solutions to the quadratic inequality are \((-∞, 3) \cup (4, ∞)\).