Question 5 of 50

2 Points

Solve the inequality 2*x*−*x*2<−8.

A. no solutions

B. −2<*x*<4

C. *x*<−2 or *x*>4

D. *x*>−2 or *x*<4 

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To solve the inequality \(2x - x^2 < -8\), let's start by rearranging it into a standard quadratic form. 

### Step-by-Step Solution:

1. **Rearrange the inequality**:

   \[
   2x - x^2 < -8
   \]

   Add \(8\) to both sides:

   \[
   -x^2 + 2x + 8 < 0
   \]

   Rearrange terms:

   \[
   -x^2 + 2x + 8 < 0
   \]

   Multiply through by \(-1\) (which reverses the inequality):

   \[
   x^2 - 2x - 8 > 0
   \]

2. **Factor the quadratic expression**:

   We need to factor \(x^2 - 2x - 8\). We look for two numbers whose product is \(-8\) and whose sum is \(-2\). These numbers are \(2\) and \(-4\).

   Therefore, the factorization is:

   \[
   (x - 4)(x + 2) > 0
   \]

3. **Determine the critical points**:

   Set each factor equal to zero to find the critical points:

   \[
   x - 4 = 0 \quad \Rightarrow \quad x = 4
   \]

   \[
   x + 2 = 0 \quad \Rightarrow \quad x = -2
   \]

   These critical points divide the real number line into intervals: \((-\infty, -2)\), \((-2, 4)\), and \((4, \infty)\).

4. **Test each interval**:

   To determine where the inequality \( (x - 4)(x + 2) > 0 \) holds true, test a value from each interval:

   - **Interval \((-\infty, -2)\)**: Choose \(x = -3\)

     \[
     (x - 4)(x + 2) = (-3 - 4)(-3 + 2) = (-7)(-1) = 7 > 0
     \]

     The inequality holds in this interval.

   - **Interval \((-2, 4)\)**: Choose \(x = 0\)

     \[
     (x - 4)(x + 2) = (0 - 4)(0 + 2) = (-4)(2) = -8 < 0
     \]

     The inequality does not hold in this interval.

   - **Interval \((4, \infty)\)**: Choose \(x = 5\)

     \[
     (x - 4)(x + 2) = (5 - 4)(5 + 2) = (1)(7) = 7 > 0
     \]

     The inequality holds in this interval.

5. **Conclusion**:

   The solution to the inequality \(x^2 - 2x - 8 > 0\) is the union of the intervals where the inequality holds:

   \[
   x < -2 \quad \text{or} \quad x > 4
   \]

   Therefore, the correct answer is:

   **C. \(x < -2\) or \(x > 4\)**

Would you like more details or have any questions? Here are some related questions:

1. How do we solve quadratic inequalities?
2. What are critical points, and how do they help in solving inequalities?
3. How can we verify solutions for quadratic inequalities?
4. What is the significance of testing intervals in inequalities?
5. How does multiplying an inequality by a negative number affect it?
6. Can we graphically represent the solution of quadratic inequalities?
7. What are some common mistakes to avoid when solving inequalities?
8. How do we deal with inequalities involving absolute values?

**Tip:** When solving quadratic inequalities, factor the quadratic expression and identify the intervals determined by the critical points to test which intervals satisfy the inequality.

Learn how to solve the quadratic inequality 2x - x^2 < -8 with a detailed step-by-step solution. This problem covers key algebraic concepts, including quadratic inequalities and factoring quadratic expressions. Suitable for students in Grades 9-12.

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