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. Specifically, we will 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 the expression inside the absolute value 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 the absolute value of $x+4$ is equal to $x+4$ itself. So, we can set up the equation $x+4=2$ and solve for $x$.

x + 4 = 2
x = -2

This means that if $x+4$ is positive, then $x=-2$.

Case 2: $x+4$ is Negative

If $x+4$ is negative, then the absolute value of $x+4$ is equal to $-(x+4)$. So, we can set up the equation $-(x+4)=2$ and solve for $x$.

-(x + 4) = 2
-x - 4 = 2
-x = 6
x = -6

This means that if $x+4$ is negative, then $x=-6$.

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 of zero.

Since we have found two solutions, $x=-2$ and $x=-6$, we can plot these points on the number line. The number line that represents the solutions to the equation $|x+4|=2$ is the number line that includes both $x=-2$ and $x=-6$.

Conclusion

In this article, we have explored the concept of absolute value equations and how to solve them. Specifically, we have focused on the equation $|x+4|=2$ and determined which number line represents the solutions to this equation. We have found that the number line that represents the solutions to the equation $|x+4|=2$ is the number line that includes both $x=-2$ and $x=-6$.

Final Answer

The final answer is the number line that includes both $x=-2$ and $x=-6$.

Introduction

In our previous article, we explored the concept of absolute value equations and how to solve them. Specifically, we 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 the expression inside the absolute value is negative.

Q: What is the difference between the two cases in solving an absolute value equation?

A: In the first case, where the expression inside the absolute value is positive, the absolute value of the expression is equal to the expression itself. In the second case, where the expression inside the absolute value is negative, the absolute value of the expression is equal to the negative of the expression.

Q: How do I determine which case to use in solving an absolute value equation?

A: To determine which case to use, you need to consider the sign of the expression inside the absolute value. If the expression is positive, use the first case. If the expression is negative, use the second case.

Q: What is the number line representation of the solutions to an absolute value equation?

A: The number line representation of the solutions to an absolute value equation is the number line that includes all the solutions to the equation.

Q: How do I plot the solutions to an absolute value equation on a number line?

A: To plot the solutions to an absolute value equation on a number line, you need to identify the solutions to the equation and plot them on the number line.

Q: What is the final answer to an absolute value equation?

A: The final answer to an absolute value equation is the number line that includes all the solutions to the equation.

Examples

Example 1: Solving the Equation $|x-3|=5$

To solve the equation $|x-3|=5$, we need to consider two cases: one where $x-3$ is positive, and one where $x-3$ is negative.

Case 1: $x-3$ is Positive

If $x-3$ is positive, then the absolute value of $x-3$ is equal to $x-3$ itself. So, we can set up the equation $x-3=5$ and solve for $x$.

x - 3 = 5
x = 8

This means that if $x-3$ is positive, then $x=8$.

Case 2: $x-3$ is Negative

If $x-3$ is negative, then the absolute value of $x-3$ is equal to $-(x-3)$. So, we can set up the equation $-(x-3)=5$ and solve for $x$.

-(x - 3) = 5
-x + 3 = 5
-x = 2
x = -2

This means that if $x-3$ is negative, then $x=-2$.

Example 2: Solving the Equation $|2x+1|=3$

To solve the equation $|2x+1|=3$, we need to consider two cases: one where $2x+1$ is positive, and one where $2x+1$ is negative.

Case 1: $2x+1$ is Positive

If $2x+1$ is positive, then the absolute value of $2x+1$ is equal to $2x+1$ itself. So, we can set up the equation $2x+1=3$ and solve for $x$.

2x + 1 = 3
2x = 2
x = 1

This means that if $2x+1$ is positive, then $x=1$.

Case 2: $2x+1$ is Negative

If $2x+1$ is negative, then the absolute value of $2x+1$ is equal to $-(2x+1)$. So, we can set up the equation $-(2x+1)=3$ and solve for $x$.

-(2x + 1) = 3
-2x - 1 = 3
-2x = 4
x = -2

This means that if $2x+1$ is negative, then $x=-2$.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about absolute value equations. We have provided examples of how to solve absolute value equations and how to plot the solutions on a number line. We have also discussed the final answer to an absolute value equation.

Final Answer

The final answer to an absolute value equation is the number line that includes all the solutions to the equation.