Determine The Solutions Of The Equation ∣ 4 X + 5 ∣ − 11 = 0 |4x + 5| - 11 = 0 ∣4 X + 5∣ − 11 = 0 .A. No Real SolutionB. 1.5 And -4C. 1.5 And 4D. 1.5

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Introduction

In this article, we will explore the solutions of the equation $|4x + 5| - 11 = 0$. The absolute value function is a fundamental concept in mathematics, and it plays a crucial role in solving equations involving absolute values. We will use the definition of the absolute value function to rewrite the equation and then solve for the variable $x$.

Understanding the Absolute Value Function

The absolute value function is defined as:

$$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

This means that if $x$ is non-negative, the absolute value of $x$ is simply $x$ itself. On the other hand, if $x$ is negative, the absolute value of $x$ is the negative of $x$.

Rewriting the Equation

Using the definition of the absolute value function, we can rewrite the equation $|4x + 5| - 11 = 0$ as:

$$|4x + 5| = 11$$

This equation states that the absolute value of $4x + 5$ is equal to 11.

Solving the Equation

To solve the equation, we need to consider two cases:

Case 1: $4x + 5 \geq 0$

In this case, the absolute value of $4x + 5$ is simply $4x + 5$ itself. Therefore, we can rewrite the equation as:

$$4x + 5 = 11$$

Subtracting 5 from both sides, we get:

$$4x = 6$$

Dividing both sides by 4, we get:

$$x = \frac{6}{4} = \frac{3}{2} = 1.5$$

Case 2: $4x + 5 < 0$

In this case, the absolute value of $4x + 5$ is the negative of $4x + 5$. Therefore, we can rewrite the equation as:

$$-(4x + 5) = 11$$

Multiplying both sides by -1, we get:

$$4x + 5 = -11$$

Subtracting 5 from both sides, we get:

$$4x = -16$$

Dividing both sides by 4, we get:

$$x = -\frac{16}{4} = -4$$

Conclusion

In conclusion, we have found two solutions to the equation $|4x + 5| - 11 = 0$: $x = 1.5$ and $x = -4$. These solutions satisfy the equation and are valid in the context of the absolute value function.

Final Answer

The final answer is: B. 1.5 and -4

Discussion

The absolute value function is a fundamental concept in mathematics, and it plays a crucial role in solving equations involving absolute values. In this article, we have used the definition of the absolute value function to rewrite the equation $|4x + 5| - 11 = 0$ and then solved for the variable $x$. We have found two solutions to the equation: $x = 1.5$ and $x = -4$. These solutions satisfy the equation and are valid in the context of the absolute value function.

Related Topics

• Absolute value function
• Solving equations involving absolute values
• Algebraic manipulations
• Mathematical reasoning

References

• [1] "Absolute Value" by Math Open Reference. Retrieved from https://www.mathopenref.com/absolutevalue.html
• [2] "Solving Equations Involving Absolute Values" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4f7/x2f1f4f8/x2f1f4f9

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.

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Introduction

In our previous article, we explored the solutions of the equation $|4x + 5| - 11 = 0$. We used the definition of the absolute value function to rewrite the equation and then solved for the variable $x$. In this article, we will answer some frequently asked questions related to the equation and its solutions.

Q&A

Q: What is the absolute value function?

A: The absolute value function is a mathematical function that returns the distance of a number from zero on the number line. It is defined as:

$$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

Q: How do I determine the solutions of an equation involving absolute values?

A: To determine the solutions of an equation involving absolute values, you need to consider two cases:

• Case 1: The expression inside the absolute value is non-negative.
• Case 2: The expression inside the absolute value is negative.

You then solve each case separately and combine the solutions to get the final answer.

Q: What are the solutions of the equation $|4x + 5| - 11 = 0$?

A: The solutions of the equation $|4x + 5| - 11 = 0$ are $x = 1.5$ and $x = -4$.

Q: Why do we need to consider two cases when solving an equation involving absolute values?

A: We need to consider two cases because the absolute value function has two different definitions depending on whether the expression inside the absolute value is non-negative or negative. By considering both cases, we can ensure that we get all possible solutions of the equation.

Q: Can I use the same method to solve other equations involving absolute values?

A: Yes, you can use the same method to solve other equations involving absolute values. The key is to identify the expression inside the absolute value and then consider the two cases.

Q: What are some common mistakes to avoid when solving equations involving absolute values?

A: Some common mistakes to avoid when solving equations involving absolute values include:

• Failing to consider both cases
• Not checking the sign of the expression inside the absolute value
• Not combining the solutions correctly

Conclusion

In conclusion, solving equations involving absolute values requires careful consideration of the two cases and attention to the sign of the expression inside the absolute value. By following the steps outlined in this article, you can determine the solutions of equations involving absolute values and avoid common mistakes.

Final Answer

The final answer is: B. 1.5 and -4

Discussion

Solving equations involving absolute values is an important skill in mathematics, and it has many applications in real-world problems. By understanding the absolute value function and how to solve equations involving absolute values, you can tackle a wide range of mathematical problems.

Related Topics

• Absolute value function
• Solving equations involving absolute values
• Algebraic manipulations
• Mathematical reasoning

References

• [1] "Absolute Value" by Math Open Reference. Retrieved from https://www.mathopenref.com/absolutevalue.html
• [2] "Solving Equations Involving Absolute Values" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4f7/x2f1f4f8/x2f1f4f9

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.