Solve The Inequality $x^2 - 2x - 15 \ \textless \ 0$.

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Introduction

In this article, we will delve into the world of inequalities and focus on solving the quadratic inequality $x^2 - 2x - 15 < 0$. This type of inequality is a fundamental concept in algebra and is used to model real-world problems. We will use various techniques to solve this inequality, including factoring, the quadratic formula, and graphing.

Understanding the Inequality

The given inequality is a quadratic inequality in the form of $ax^2 + bx + c < 0$, where $a = 1$, $b = -2$, and $c = -15$. To solve this inequality, we need to find the values of $x$ that make the quadratic expression $x^2 - 2x - 15$ less than zero.

Factoring the Quadratic Expression

One way to solve this inequality is to factor the quadratic expression $x^2 - 2x - 15$. We can factor this expression as $(x - 5)(x + 3)$. This means that the inequality can be rewritten as $(x - 5)(x + 3) < 0$.

Finding the Critical Points

To solve the inequality, we need to find the critical points, which are the values of $x$ that make the quadratic expression equal to zero. In this case, the critical points are $x = 5$ and $x = -3$. These points divide the number line into three intervals: $(-\infty, -3)$, $(-3, 5)$, and $(5, \infty)$.

Testing the Intervals

To determine which intervals satisfy the inequality, we can test a value from each interval. Let's choose $x = -4$ from the interval $(-\infty, -3)$, $x = 0$ from the interval $(-3, 5)$, and $x = 6$ from the interval $(5, \infty)$. Plugging these values into the inequality, we get:

• For $x = -4$: $(-4 - 5)(-4 + 3) = (-9)(-1) = 9 > 0$
• For $x = 0$: $(0 - 5)(0 + 3) = (-5)(3) = -15 < 0$
• For $x = 6$: $(6 - 5)(6 + 3) = (1)(9) = 9 > 0$

Determining the Solution Set

Based on the test values, we can determine that the interval $(-3, 5)$ satisfies the inequality. This means that the solution set is $x \in (-3, 5)$.

Graphing the Solution Set

To visualize the solution set, we can graph the quadratic function $y = x^2 - 2x - 15$ and shade the region below the curve. The graph shows that the solution set is the interval $(-3, 5)$.

Conclusion

In this article, we solved the quadratic inequality $x^2 - 2x - 15 < 0$ using various techniques, including factoring, the quadratic formula, and graphing. We found that the solution set is $x \in (-3, 5)$, which can be represented graphically as the region below the curve of the quadratic function.

Final Thoughts

Solving quadratic inequalities is an essential skill in algebra and is used to model real-world problems. By understanding the concepts and techniques presented in this article, you can solve a wide range of quadratic inequalities and apply them to various fields, such as physics, engineering, and economics.

Additional Resources

For further practice and review, you can try solving the following quadratic inequalities:

• $x^2 + 4x + 4 < 0$
• $x^2 - 6x + 8 < 0$
• $x^2 + 2x - 6 < 0$

You can also explore online resources, such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha, to learn more about quadratic inequalities and other mathematical concepts.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman

Note: The references provided are for general mathematical resources and are not specific to the topic of quadratic inequalities.

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Introduction

In our previous article, we solved the quadratic inequality $x^2 - 2x - 15 < 0$ using various techniques, including factoring, the quadratic formula, and graphing. In this article, we will address some common questions and concerns that readers may have about solving quadratic inequalities.

Q&A

Q: What is the difference between a quadratic equation and a quadratic inequality?

A: A quadratic equation is an equation in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. A quadratic inequality, on the other hand, is an inequality in the form of $ax^2 + bx + c < 0$, $ax^2 + bx + c > 0$, or $ax^2 + bx + c = 0$. The main difference between the two is that a quadratic equation has a specific solution, while a quadratic inequality has a range of solutions.

Q: How do I know which technique to use when solving a quadratic inequality?

A: The choice of technique depends on the specific inequality and the values of $a$, $b$, and $c$. If the inequality can be factored easily, factoring may be the best approach. If the inequality cannot be factored easily, the quadratic formula may be used. Graphing can also be a useful technique, especially when visualizing the solution set.

Q: What is the significance of the critical points in solving a quadratic inequality?

A: The critical points are the values of $x$ that make the quadratic expression equal to zero. These points divide the number line into intervals, and the solution set is determined by testing a value from each interval.

Q: Can I use the quadratic formula to solve a quadratic inequality?

A: Yes, the quadratic formula can be used to solve a quadratic inequality. However, it is often more efficient to use factoring or graphing when possible.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use a graphing calculator or software, or you can plot points on a coordinate plane. The graph of a quadratic function is a parabola, and the solution set is the region below the curve.

Q: What is the relationship between the solution set of a quadratic inequality and the graph of the quadratic function?

A: The solution set of a quadratic inequality is the region below the curve of the quadratic function. This means that if the quadratic function is above the x-axis, the solution set is empty. If the quadratic function is below the x-axis, the solution set is the entire number line.

Q: Can I use technology to solve quadratic inequalities?

A: Yes, technology can be a useful tool in solving quadratic inequalities. Graphing calculators and software can help you visualize the solution set and make it easier to identify the intervals that satisfy the inequality.

Q: What are some common mistakes to avoid when solving quadratic inequalities?

A: Some common mistakes to avoid include:

• Not factoring the quadratic expression correctly
• Not identifying the critical points correctly
• Not testing a value from each interval
• Not considering the sign of the quadratic expression in each interval

Conclusion

Solving quadratic inequalities can be a challenging task, but with practice and patience, you can become proficient in using various techniques to solve these types of inequalities. By understanding the concepts and techniques presented in this article, you can solve a wide range of quadratic inequalities and apply them to various fields, such as physics, engineering, and economics.

Final Thoughts

Quadratic inequalities are an essential part of algebra and are used to model real-world problems. By mastering the techniques and concepts presented in this article, you can solve a wide range of quadratic inequalities and apply them to various fields.

Additional Resources

For further practice and review, you can try solving the following quadratic inequalities:

• $x^2 + 4x + 4 < 0$
• $x^2 - 6x + 8 < 0$
• $x^2 + 2x - 6 < 0$

You can also explore online resources, such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha, to learn more about quadratic inequalities and other mathematical concepts.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman