Which Values Are Solutions To The Inequality Below? Check All That Apply. X 2 \textless 16 X^2 \ \textless \ 16 X 2 \textless 16 A. 3 B. 5 C. 4 D. -1

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Introduction

Quadratic inequalities are a fundamental concept in algebra, and solving them requires a deep understanding of quadratic equations and their properties. In this article, we will explore the solution to the inequality x2<16x^2 < 16 and determine which values are solutions to this inequality.

Understanding the Inequality

The given inequality is x2<16x^2 < 16. This is a quadratic inequality, where the variable xx is squared and the result is less than 16. To solve this inequality, we need to find the values of xx that satisfy this condition.

Solving the Inequality

To solve the inequality x2<16x^2 < 16, we can start by finding the square root of both sides. However, since we are dealing with an inequality, we need to consider both the positive and negative square roots.

Finding the Square Root

The square root of 16 is 4, since 42=164^2 = 16. However, we also need to consider the negative square root, which is -4, since (−4)2=16(-4)^2 = 16.

Solving for x

Now that we have found the square roots of 16, we can solve for xx. We have two cases to consider:

  • x2<16x^2 < 16 implies that x<4x < 4 or x>−4x > -4.
  • However, since we are dealing with a quadratic inequality, we need to consider the sign of the expression x2−16x^2 - 16.

Analyzing the Sign of the Expression

The expression x2−16x^2 - 16 is a quadratic expression that can be factored as (x−4)(x+4)(x - 4)(x + 4). This expression is negative when −4<x<4-4 < x < 4.

Combining the Results

Combining the results from the previous steps, we can conclude that the solution to the inequality x2<16x^2 < 16 is −4<x<4-4 < x < 4.

Checking the Solutions

Now that we have found the solution to the inequality, we can check which values are solutions to this inequality. The given options are:

  • A. 3
  • B. 5
  • C. 4
  • D. -1

We can plug each of these values into the inequality x2<16x^2 < 16 to check if they are solutions.

Checking Option A

Plugging in x=3x = 3 into the inequality x2<16x^2 < 16, we get 32<163^2 < 16, which is true.

Checking Option B

Plugging in x=5x = 5 into the inequality x2<16x^2 < 16, we get 52<165^2 < 16, which is false.

Checking Option C

Plugging in x=4x = 4 into the inequality x2<16x^2 < 16, we get 42<164^2 < 16, which is false.

Checking Option D

Plugging in x=−1x = -1 into the inequality x2<16x^2 < 16, we get (−1)2<16(-1)^2 < 16, which is true.

Conclusion

In conclusion, the solution to the inequality x2<16x^2 < 16 is −4<x<4-4 < x < 4. We can check which values are solutions to this inequality by plugging them into the inequality. The values that satisfy the inequality are:

  • A. 3
  • D. -1

These values are solutions to the inequality x2<16x^2 < 16.

Frequently Asked Questions

Q: What is the solution to the inequality x2<16x^2 < 16?

A: The solution to the inequality x2<16x^2 < 16 is −4<x<4-4 < x < 4.

Q: How do I check if a value is a solution to the inequality x2<16x^2 < 16?

A: To check if a value is a solution to the inequality x2<16x^2 < 16, plug the value into the inequality and check if it is true.

Q: What are the values that satisfy the inequality x2<16x^2 < 16?

A: The values that satisfy the inequality x2<16x^2 < 16 are −4<x<4-4 < x < 4. Specifically, the values that are solutions to this inequality are:

  • A. 3
  • D. -1

Final Thoughts

Solving quadratic inequalities requires a deep understanding of quadratic equations and their properties. By following the steps outlined in this article, we can solve the inequality x2<16x^2 < 16 and determine which values are solutions to this inequality. The values that satisfy the inequality are −4<x<4-4 < x < 4, and specifically, the values that are solutions to this inequality are:

  • A. 3
  • D. -1

I hope this article has provided a clear and concise guide to solving quadratic inequalities. If you have any questions or need further clarification, please don't hesitate to ask.

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Introduction

In our previous article, we explored the solution to the inequality x2<16x^2 < 16 and determined which values are solutions to this inequality. In this article, we will continue to provide a Q&A guide to help you better understand quadratic inequalities and their solutions.

Q&A Guide

Q: What is a quadratic inequality?

A: A quadratic inequality is an inequality that involves a quadratic expression, such as x2<16x^2 < 16.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to find the values of xx that satisfy the inequality. This can be done by finding the square root of both sides of the inequality and considering both the positive and negative square roots.

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

A: A quadratic equation is an equation that involves a quadratic expression, such as x2=16x^2 = 16. A quadratic inequality, on the other hand, is an inequality that involves a quadratic expression, such as x2<16x^2 < 16.

Q: How do I determine the solution to a quadratic inequality?

A: To determine the solution to a quadratic inequality, you need to consider the sign of the expression x2−16x^2 - 16. If the expression is negative, then the inequality is true. If the expression is positive, then the inequality is false.

Q: What are some common quadratic inequalities?

A: Some common quadratic inequalities include:

  • x2<16x^2 < 16
  • x2>16x^2 > 16
  • x2=16x^2 = 16

Q: How do I check if a value is a solution to a quadratic inequality?

A: To check if a value is a solution to a quadratic inequality, plug the value into the inequality and check if it is true.

Q: What are some tips for solving quadratic inequalities?

A: Some tips for solving quadratic inequalities include:

  • Make sure to consider both the positive and negative square roots.
  • Use a number line to visualize the solution to the inequality.
  • Check your work by plugging in values into the inequality.

Common Quadratic Inequalities

x2<16x^2 < 16

The solution to the inequality x2<16x^2 < 16 is −4<x<4-4 < x < 4. Specifically, the values that are solutions to this inequality are:

  • A. 3
  • D. -1

x2>16x^2 > 16

The solution to the inequality x2>16x^2 > 16 is x<−4x < -4 or x>4x > 4. Specifically, the values that are solutions to this inequality are:

  • A. 3
  • B. 5
  • C. 4

x2=16x^2 = 16

The solution to the inequality x2=16x^2 = 16 is x=−4x = -4 or x=4x = 4. Specifically, the values that are solutions to this inequality are:

  • C. 4
  • D. -1

Conclusion

In conclusion, quadratic inequalities are an important concept in algebra, and solving them requires a deep understanding of quadratic equations and their properties. By following the steps outlined in this article, you can solve quadratic inequalities and determine which values are solutions to these inequalities. Remember to consider both the positive and negative square roots, use a number line to visualize the solution, and check your work by plugging in values into the inequality.

Frequently Asked Questions

Q: What is a quadratic inequality?

A: A quadratic inequality is an inequality that involves a quadratic expression, such as x2<16x^2 < 16.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to find the values of xx that satisfy the inequality. This can be done by finding the square root of both sides of the inequality and considering both the positive and negative square roots.

Q: What are some common quadratic inequalities?

A: Some common quadratic inequalities include:

  • x2<16x^2 < 16
  • x2>16x^2 > 16
  • x2=16x^2 = 16

Q: How do I check if a value is a solution to a quadratic inequality?

A: To check if a value is a solution to a quadratic inequality, plug the value into the inequality and check if it is true.

Final Thoughts

Solving quadratic inequalities requires a deep understanding of quadratic equations and their properties. By following the steps outlined in this article, you can solve quadratic inequalities and determine which values are solutions to these inequalities. Remember to consider both the positive and negative square roots, use a number line to visualize the solution, and check your work by plugging in values into the inequality.