Solve The Inequality Algebraically:${ X^3 - 2x^2 - 80x \ \textgreater \ 0 }$The Solution Is { \square$}$.(Simplify Your Answer. Type Your Answer In Interval Notation.)

Mar 11, 2025 by ADMIN 171 views

**Solving Inequalities Algebraically: A Step-by-Step Guide**

Introduction

In algebra, inequalities are mathematical statements that compare two expressions, indicating that one is greater than, less than, or equal to the other. Solving inequalities algebraically involves manipulating the inequality to isolate the variable and find the solution set. In this article, we will focus on solving the inequality $x^3 - 2x^2 - 80x > 0$ algebraically.

Understanding the Inequality

The given inequality is a cubic inequality, which means it involves a cubic expression. The expression $x^3 - 2x^2 - 80x$ can be factored as $x(x^2 - 2x - 80)$ . To solve the inequality, we need to find the values of $x$ that make the expression positive.

Factoring the Expression

To factor the expression $x^2 - 2x - 80$ , we need to find two numbers whose product is $-80$ and whose sum is $-2$ . These numbers are $-10$ and $8$ , so we can write the expression as $(x - 10)(x + 8)$ .

Solving the Inequality

Now that we have factored the expression, we can rewrite the inequality as $x(x - 10)(x + 8) > 0$ . To solve this inequality, we need to find the values of $x$ that make the expression positive.

Finding the Critical Points

The critical points are the values of $x$ that make the expression equal to zero. In this case, the critical points are $x = 0$ , $x = 10$ , and $x = -8$ .

Creating a Sign Chart

To determine the solution set, we need to create a sign chart. The sign chart shows the sign of the expression in each interval between the critical points.

Interval	Sign of $x$	Sign of $(x - 10)$	Sign of $(x + 8)$	Sign of $x(x - 10)(x + 8)$
$(-\infty, -8)$	-	-	-	+
$(-8, 0)$	-	-	+	-
$(0, 10)$	+	-	+	-
$(10, \infty)$	+	+	+	+

Determining the Solution Set

From the sign chart, we can see that the expression is positive when $x < -8$ , $x > 10$ , and $x = 0$ . Therefore, the solution set is $(-\infty, -8) \cup (0, 10) \cup (10, \infty)$ .

Conclusion

Solving inequalities algebraically involves manipulating the inequality to isolate the variable and find the solution set. In this article, we solved the inequality $x^3 - 2x^2 - 80x > 0$ algebraically by factoring the expression and creating a sign chart. The solution set is $(-\infty, -8) \cup (0, 10) \cup (10, \infty)$ .

Key Takeaways

To solve a cubic inequality, we need to factor the expression and find the critical points.
A sign chart can be used to determine the solution set by showing the sign of the expression in each interval between the critical points.
The solution set can be expressed in interval notation.

Common Mistakes to Avoid

Not factoring the expression correctly.
Not finding all the critical points.
Not creating a sign chart to determine the solution set.

Real-World Applications

Solving inequalities algebraically has many real-world applications, such as:

Modeling population growth and decline.
Determining the maximum and minimum values of a function.
Finding the solution to a system of linear equations.

Practice Problems

Solve the inequality $x^2 + 5x + 6 > 0$ algebraically.
Solve the inequality $x^3 - 3x^2 - 4x > 0$ algebraically.
Solve the inequality $x^2 - 2x - 15 > 0$ algebraically.

Conclusion

Introduction

In our previous article, we discussed how to solve inequalities algebraically by factoring the expression and creating a sign chart. In this article, we will answer some common questions that students often have when solving inequalities algebraically.

Q: What is the first step in solving an inequality algebraically?

A: The first step in solving an inequality algebraically is to simplify the inequality by combining like terms and isolating the variable.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. These numbers are called the factors of the quadratic expression.

Q: What is a sign chart, and how do I use it to solve an inequality?

A: A sign chart is a table that shows the sign of the expression in each interval between the critical points. To use a sign chart to solve an inequality, you need to identify the critical points, create a sign chart, and determine the solution set based on the sign of the expression in each interval.

Q: How do I determine the solution set of an inequality?

A: To determine the solution set of an inequality, you need to identify the intervals where the expression is positive or negative, and then express the solution set in interval notation.

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

A: Some common mistakes to avoid when solving inequalities algebraically include:

Not factoring the expression correctly.
Not finding all the critical points.
Not creating a sign chart to determine the solution set.
Not expressing the solution set in interval notation.

Q: How do I apply the concept of solving inequalities algebraically to real-world problems?

A: The concept of solving inequalities algebraically can be applied to real-world problems such as:

Modeling population growth and decline.
Determining the maximum and minimum values of a function.
Finding the solution to a system of linear equations.

Q: What are some tips for solving inequalities algebraically?

A: Some tips for solving inequalities algebraically include:

Simplify the inequality by combining like terms and isolating the variable.
Factor the expression to find the critical points.
Create a sign chart to determine the solution set.
Express the solution set in interval notation.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you need to substitute the values of the variable into the original inequality and verify that the inequality is true.

Q: What are some common types of inequalities that I may encounter?

A: Some common types of inequalities that you may encounter include:

Linear inequalities: inequalities that involve a linear expression.
Quadratic inequalities: inequalities that involve a quadratic expression.
Polynomial inequalities: inequalities that involve a polynomial expression.

Conclusion

Solving inequalities algebraically is an important skill in algebra that has many real-world applications. By following the steps outlined in this article, you can solve inequalities algebraically and express the solution set in interval notation. Remember to simplify the inequality, factor the expression, create a sign chart, and express the solution set in interval notation.

Practice Problems

Solve the inequality $x^2 + 5x + 6 > 0$ algebraically.
Solve the inequality $x^3 - 3x^2 - 4x > 0$ algebraically.
Solve the inequality $x^2 - 2x - 15 > 0$ algebraically.

Additional Resources

For more information on solving inequalities algebraically, you can refer to the following resources:

Algebra textbooks: many algebra textbooks include chapters on solving inequalities algebraically.
Online resources: there are many online resources available that provide step-by-step instructions on solving inequalities algebraically.
Video tutorials: video tutorials can provide a visual explanation of how to solve inequalities algebraically.