Which Number Line Represents The Solutions To ∣ X + 4 ∣ = 2 |x+4|=2 ∣ X + 4∣ = 2 ?

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Introduction

In mathematics, absolute value equations are a fundamental concept that deals with the distance of a number from zero on the number line. The absolute value of a number is its distance from zero, and it is always non-negative. In this article, we will explore the concept of absolute value equations and how to solve them. We will specifically focus on the equation $|x+4|=2$ and determine which number line represents the solutions to this equation.

Understanding Absolute Value Equations

Absolute value equations are equations that involve the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5. This is because both 5 and -5 are 5 units away from zero on the number line.

When solving absolute value equations, we need to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative. This is because the absolute value of a number is always non-negative, so we need to consider both the positive and negative possibilities.

Solving the Equation $|x+4|=2$

To solve the equation $|x+4|=2$, we need to consider two cases: one where $x+4$ is positive, and one where $x+4$ is negative.

Case 1: $x+4$ is Positive

If $x+4$ is positive, then we can write the equation as $x+4=2$. To solve for $x$, we need to isolate the variable $x$. We can do this by subtracting 4 from both sides of the equation:

$$x+4=2$$

$$x=2-4$$

$$x=-2$$

So, in this case, the solution to the equation is $x=-2$.

Case 2: $x+4$ is Negative

If $x+4$ is negative, then we can write the equation as $x+4=-2$. To solve for $x$, we need to isolate the variable $x$. We can do this by subtracting 4 from both sides of the equation:

$$x+4=-2$$

$$x=-2-4$$

$$x=-6$$

So, in this case, the solution to the equation is $x=-6$.

Determining the Number Line Representation

Now that we have found the solutions to the equation $|x+4|=2$, we need to determine which number line represents these solutions. The number line is a visual representation of the real numbers, with positive numbers to the right of zero and negative numbers to the left.

The solution $x=-2$ represents a point on the number line that is 2 units to the left of zero. Similarly, the solution $x=-6$ represents a point on the number line that is 6 units to the left of zero.

Conclusion

In conclusion, the number line that represents the solutions to the equation $|x+4|=2$ is a number line with two points: one at $x=-2$ and one at $x=-6$. These points are 2 and 6 units to the left of zero, respectively.

Final Answer

The final answer is a number line with two points: one at $x=-2$ and one at $x=-6$.

Introduction

In our previous article, we explored the concept of absolute value equations and how to solve them. We specifically focused on the equation $|x+4|=2$ and determined which number line represents the solutions to this equation. In this article, we will answer some frequently asked questions (FAQs) about absolute value equations.

Q&A

Q: What is an absolute value equation?

A: An absolute value equation is an equation that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative. You can then solve each case separately to find the solutions to the equation.

Q: What is the difference between a positive and negative absolute value equation?

A: A positive absolute value equation is one where the expression inside the absolute value is positive, while a negative absolute value equation is one where the expression inside the absolute value is negative. For example, $|x+4|=2$ is a positive absolute value equation, while $|x+4|=-2$ is a negative absolute value equation.

Q: How do I determine which number line represents the solutions to an absolute value equation?

A: To determine which number line represents the solutions to an absolute value equation, you need to consider the solutions to the equation. The number line is a visual representation of the real numbers, with positive numbers to the right of zero and negative numbers to the left.

Q: Can I use a calculator to solve an absolute value equation?

A: Yes, you can use a calculator to solve an absolute value equation. However, it's always a good idea to check your work by hand to make sure you understand the solution.

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

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

• Not considering both the positive and negative cases
• Not isolating the variable correctly
• Not checking your work by hand

Q: Can I use absolute value equations to solve real-world problems?

A: Yes, absolute value equations can be used to solve real-world problems. For example, you can use absolute value equations to model the distance between two points on a number line.

Q: What are some advanced topics related to absolute value equations?

A: Some advanced topics related to absolute value equations include:

• Solving absolute value inequalities
• Graphing absolute value functions
• Using absolute value equations to model real-world problems

Conclusion

In conclusion, absolute value equations are a fundamental concept in mathematics that deals with the distance of a number from zero on the number line. By understanding how to solve absolute value equations, you can apply this knowledge to solve real-world problems and model complex situations.

Final Answer

The final answer is that absolute value equations are a powerful tool for solving real-world problems and modeling complex situations. By understanding how to solve absolute value equations, you can apply this knowledge to a wide range of fields, including science, engineering, and finance.

Additional Resources

• Khan Academy: Absolute Value Equations
• Mathway: Absolute Value Equations
• Wolfram Alpha: Absolute Value Equations

Note: The above resources are provided for additional learning and practice. They are not a substitute for the information provided in this article.