Solve The Inequality:${ 2 \sqrt[3]{x} - 5 \geq 3 }$
Introduction
In this article, we will delve into the world of inequalities and learn how to solve a specific one involving a cube root. The given inequality is 2\sqrt[3]{x} - 5 ≥ 3, and our goal is to isolate the variable x and find the solution set. We will break down the steps involved in solving this inequality and provide a clear explanation of each step.
Understanding the Inequality
The given inequality is 2\sqrt[3]{x} - 5 ≥ 3. This means that the expression 2\sqrt[3]{x} - 5 is greater than or equal to 3. Our objective is to solve for x, which means we need to isolate x on one side of the inequality.
Step 1: Add 5 to Both Sides
To start solving the inequality, we need to get rid of the constant term -5 on the left-hand side. We can do this by adding 5 to both sides of the inequality. This will give us:
2\sqrt[3]{x} ≥ 3 + 5
Step 2: Simplify the Right-Hand Side
Now, we can simplify the right-hand side by combining the constants:
2\sqrt[3]{x} ≥ 8
Step 3: Divide Both Sides by 2
Next, we need to get rid of the coefficient 2 on the left-hand side. We can do this by dividing both sides of the inequality by 2. This will give us:
\sqrt[3]{x} ≥ 4
Step 4: Cube Both Sides
To isolate x, we need to get rid of the cube root on the left-hand side. We can do this by cubing both sides of the inequality. This will give us:
x ≥ 4^3
Step 5: Simplify the Right-Hand Side
Now, we can simplify the right-hand side by cubing the constant:
x ≥ 64
Conclusion
In this article, we have solved the inequality 2\sqrt[3]{x} - 5 ≥ 3. We broke down the steps involved in solving this inequality and provided a clear explanation of each step. The final solution is x ≥ 64, which means that x is greater than or equal to 64.
Tips and Tricks
- When solving inequalities involving cube roots, it's essential to remember that the cube root function is increasing, meaning that as the input increases, the output also increases.
- When cubing both sides of an inequality, make sure to check for extraneous solutions. In this case, we didn't have any extraneous solutions, but it's always a good idea to check.
Real-World Applications
Solving inequalities involving cube roots has many real-world applications, such as:
- Modeling population growth: The cube root function can be used to model population growth, where the input represents the number of individuals and the output represents the population size.
- Analyzing data: The cube root function can be used to analyze data that follows a cubic distribution, such as the distribution of particle sizes in a sample.
Final Thoughts
Solving inequalities involving cube roots requires a clear understanding of the properties of the cube root function and the steps involved in solving inequalities. By following the steps outlined in this article, you can solve inequalities involving cube roots and apply them to real-world problems.
Common Mistakes to Avoid
- When solving inequalities involving cube roots, it's essential to remember that the cube root function is increasing, meaning that as the input increases, the output also increases.
- When cubing both sides of an inequality, make sure to check for extraneous solutions. In this case, we didn't have any extraneous solutions, but it's always a good idea to check.
Additional Resources
- For more information on solving inequalities involving cube roots, check out the following resources:
- Khan Academy: Solving Inequalities Involving Cube Roots
- Mathway: Solving Inequalities Involving Cube Roots
- Wolfram Alpha: Solving Inequalities Involving Cube Roots
Conclusion
In conclusion, solving the inequality 2\sqrt[3]{x} - 5 ≥ 3 requires a clear understanding of the properties of the cube root function and the steps involved in solving inequalities. By following the steps outlined in this article, you can solve inequalities involving cube roots and apply them to real-world problems.
Introduction
In our previous article, we solved the inequality 2\sqrt[3]{x} - 5 ≥ 3 and found that the solution is x ≥ 64. However, we know that there are many more questions and doubts that our readers may have. In this article, we will address some of the most frequently asked questions and provide additional information to help you better understand the concept of solving inequalities involving cube roots.
Q&A
Q: What is the difference between a cube root and a square root?
A: A cube root is a mathematical operation that finds the value that, when cubed, gives a specified value. For example, the cube root of 64 is 4, because 4^3 = 64. On the other hand, a square root is a mathematical operation that finds the value that, when squared, gives a specified value. For example, the square root of 16 is 4, because 4^2 = 16.
Q: How do I know which side of the inequality to add or subtract from?
A: When solving an inequality, you need to add or subtract the same value from both sides to maintain the inequality. For example, if you have the inequality 2\sqrt[3]{x} - 5 ≥ 3, you can add 5 to both sides to get 2\sqrt[3]{x} ≥ 8.
Q: What is the difference between a cube root and a power of 3?
A: A cube root is a mathematical operation that finds the value that, when cubed, gives a specified value. On the other hand, a power of 3 is a mathematical operation that raises a number to the power of 3. For example, 3^3 = 27, but the cube root of 27 is 3.
Q: Can I use the same steps to solve inequalities involving square roots?
A: No, the steps to solve inequalities involving square roots are different from those involving cube roots. When solving inequalities involving square roots, you need to use the square root symbol (√) and follow the rules of square root operations.
Q: How do I know if I have an extraneous solution?
A: An extraneous solution is a solution that is not valid for the original inequality. To check for extraneous solutions, you need to plug the solution back into the original inequality and check if it is true. If it is not true, then the solution is extraneous.
Q: Can I use a calculator to solve inequalities involving cube roots?
A: Yes, you can use a calculator to solve inequalities involving cube roots. However, you need to make sure that the calculator is set to the correct mode (e.g., cube root mode) and that you are using the correct operation (e.g., cubing).
Tips and Tricks
- When solving inequalities involving cube roots, make sure to check for extraneous solutions.
- When using a calculator to solve inequalities involving cube roots, make sure to set the calculator to the correct mode and use the correct operation.
- When solving inequalities involving cube roots, it's essential to remember that the cube root function is increasing, meaning that as the input increases, the output also increases.
Real-World Applications
Solving inequalities involving cube roots has many real-world applications, such as:
- Modeling population growth: The cube root function can be used to model population growth, where the input represents the number of individuals and the output represents the population size.
- Analyzing data: The cube root function can be used to analyze data that follows a cubic distribution, such as the distribution of particle sizes in a sample.
Final Thoughts
Solving inequalities involving cube roots requires a clear understanding of the properties of the cube root function and the steps involved in solving inequalities. By following the steps outlined in this article and using the tips and tricks provided, you can solve inequalities involving cube roots and apply them to real-world problems.
Common Mistakes to Avoid
- When solving inequalities involving cube roots, make sure to check for extraneous solutions.
- When using a calculator to solve inequalities involving cube roots, make sure to set the calculator to the correct mode and use the correct operation.
- When solving inequalities involving cube roots, it's essential to remember that the cube root function is increasing, meaning that as the input increases, the output also increases.
Additional Resources
- For more information on solving inequalities involving cube roots, check out the following resources:
- Khan Academy: Solving Inequalities Involving Cube Roots
- Mathway: Solving Inequalities Involving Cube Roots
- Wolfram Alpha: Solving Inequalities Involving Cube Roots
Conclusion
In conclusion, solving the inequality 2\sqrt[3]{x} - 5 ≥ 3 requires a clear understanding of the properties of the cube root function and the steps involved in solving inequalities. By following the steps outlined in this article and using the tips and tricks provided, you can solve inequalities involving cube roots and apply them to real-world problems.