Solve the inequalities 1.x²-5x-14>0         2.x²-5x-14<0

To solve the inequalities:

### 1. \( x^2 - 5x - 14 > 0 \)
### 2. \( x^2 - 5x - 14 < 0 \)

We start by finding the roots of the equation \( x^2 - 5x - 14 = 0 \), as these will help determine the solution intervals for both inequalities.

#### Step 1: Solve \( x^2 - 5x - 14 = 0 \) using the quadratic formula
The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( x^2 - 5x - 14 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -5 \)
- \( c = -14 \)

Now, apply the quadratic formula:

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)}
\]

\[
x = \frac{5 \pm \sqrt{25 + 56}}{2}
\]

\[
x = \frac{5 \pm \sqrt{81}}{2}
\]

\[
x = \frac{5 \pm 9}{2}
\]

So, the two solutions are:

\[
x = \frac{5 + 9}{2} = 7 \quad \text{and} \quad x = \frac{5 - 9}{2} = -2
\]

Thus, the roots are \( x = 7 \) and \( x = -2 \).

#### Step 2: Analyze the intervals
The solutions divide the number line into three intervals:
- \( (-\infty, -2) \)
- \( (-2, 7) \)
- \( (7, \infty) \)

We will now test the sign of \( x^2 - 5x - 14 \) in each interval to determine where it is greater than 0 and where it is less than 0.

1. **For \( x^2 - 5x - 14 > 0 \):**
   We need the expression to be positive. Test each interval:

   - In the interval \( (-\infty, -2) \), pick a test point like \( x = -3 \):
     \[
     (-3)^2 - 5(-3) - 14 = 9 + 15 - 14 = 10 > 0
     \]
     So, the expression is positive in \( (-\infty, -2) \).

   - In the interval \( (-2, 7) \), pick a test point like \( x = 0 \):
     \[
     0^2 - 5(0) - 14 = -14 < 0
     \]
     So, the expression is negative in \( (-2, 7) \).

   - In the interval \( (7, \infty) \), pick a test point like \( x = 8 \):
     \[
     8^2 - 5(8) - 14 = 64 - 40 - 14 = 10 > 0
     \]
     So, the expression is positive in \( (7, \infty) \).

   Therefore, the solution to \( x^2 - 5x - 14 > 0 \) is:
   \[
   x \in (-\infty, -2) \cup (7, \infty)
   \]

2. **For \( x^2 - 5x - 14 < 0 \):**
   We need the expression to be negative. From the tests above, we know the expression is negative in \( (-2, 7) \).

   Therefore, the solution to \( x^2 - 5x - 14 < 0 \) is:
   \[
   x \in (-2, 7)
   \]

### Final Solutions:
1. \( x^2 - 5x - 14 > 0 \) gives \( x \in (-\infty, -2) \cup (7, \infty) \)
2. \( x^2 - 5x - 14 < 0 \) gives \( x \in (-2, 7) \)

Would you like more details on how I tested each interval, or any further explanation?

### Related Questions:
1. What happens if the inequality was \( x^2 - 5x - 14 \geq 0 \)?
2. How does the quadratic formula handle non-real roots?
3. Can a quadratic inequality have no real solution?
4. What are the practical applications of solving inequalities?
5. How do inequalities differ from equalities in geometric terms?

### Tip:
Always test points from each interval when solving inequalities; they help confirm the sign of the expression.

Solve the inequalities 1.x²-5x-14>0 2.x²-5x-14<0

This guide provides a step-by-step solution to the quadratic inequalities x² - 5x - 14 > 0 and x² - 5x - 14 < 0. Learn how to apply the quadratic formula and analyze intervals for inequality solutions. Suitable for students in Grades 8-10.

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