find the set of values of x that satisfy the inequality 9x^2-15x<6

To solve the inequality \(9x^2 - 15x < 6\), follow these steps:

1. **Rewrite the inequality in standard form**:
   \[
   9x^2 - 15x - 6 < 0
   \]

2. **Find the roots of the corresponding equation**:
   Solve the quadratic equation:
   \[
   9x^2 - 15x - 6 = 0
   \]
   We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 9\), \(b = -15\), and \(c = -6\).

   \[
   x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4(9)(-6)}}{2(9)}
   \]
   \[
   x = \frac{15 \pm \sqrt{225 + 216}}{18}
   \]
   \[
   x = \frac{15 \pm \sqrt{441}}{18}
   \]
   \[
   x = \frac{15 \pm 21}{18}
   \]

   The roots are:
   \[
   x_1 = \frac{36}{18} = 2, \quad x_2 = \frac{-6}{18} = -\frac{1}{3}
   \]

3. **Determine the sign of the quadratic expression**:
   The roots \(x = 2\) and \(x = -\frac{1}{3}\) divide the number line into three intervals:
   - \(x < -\frac{1}{3}\)
   - \(-\frac{1}{3} < x < 2\)
   - \(x > 2\)

   Choose a test point in each interval to determine the sign of the quadratic expression \(9x^2 - 15x - 6\):

   - For \(x < -\frac{1}{3}\) (e.g., \(x = -1\)):
     \[
     9(-1)^2 - 15(-1) - 6 = 9 + 15 - 6 = 18 \quad (\text{positive})
     \]
   - For \(-\frac{1}{3} < x < 2\) (e.g., \(x = 0\)):
     \[
     9(0)^2 - 15(0) - 6 = -6 \quad (\text{negative})
     \]
   - For \(x > 2\) (e.g., \(x = 3\)):
     \[
     9(3)^2 - 15(3) - 6 = 81 - 45 - 6 = 30 \quad (\text{positive})
     \]

4. **Conclusion**:
   The quadratic expression is negative between the roots \(x = -\frac{1}{3}\) and \(x = 2\). Therefore, the solution to the inequality is:
   \[
   -\frac{1}{3} < x < 2
   \]

So, the set of values of \(x\) that satisfy the inequality \(9x^2 - 15x < 6\) is \(\boxed{-\frac{1}{3} < x < 2}\).

Would you like further details or have any questions?

---

Here are some related questions:

1. How do you solve quadratic inequalities in general?
2. What are the steps to apply the quadratic formula?
3. How do you determine the sign of a quadratic expression on different intervals?
4. How do you check the solution to a quadratic inequality?
5. Can you explain the concept of test points in interval testing?
6. What is the significance of the discriminant in quadratic equations?
7. How do you graph quadratic inequalities?
8. What is the vertex form of a quadratic equation, and how is it useful?

**Tip**: Always verify your solution by testing points in each interval defined by the roots to ensure the correct sign of the quadratic expression.

Learn how to solve the quadratic inequality 9x^2 - 15x < 6 step-by-step with detailed explanations and interval testing. This problem involves quadratic equations and inequalities, suitable for advanced high school students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation