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

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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.

To solve an absolute value equation, 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 we can write the equation as $x+4=2$. To solve for $x$, we need to isolate $x$ on one side of the equation. We can do this by subtracting 4 from both sides of the equation:

$x+4-4=2-4$

$x=2-4$

$x=-2$

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 $x$ on one side of the equation. We can do this by distributing the negative sign to the expression inside the parentheses:

$-x-4=2$

To get rid of the negative sign, we can multiply both sides of the equation by -1:

$-(-x-4)=-2$

$x+4=-2$

Now, we can isolate $x$ by subtracting 4 from both sides of the equation:

$x+4-4=-2-4$

$x=-6$

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 solutions to an equation, and it can help us understand the relationship between the solutions.

To determine the number line representation, we need to plot the solutions on the number line. The number line has a scale that represents the distance of a number from zero. We can plot the solutions by marking the points on the number line that correspond to the solutions we found earlier.

For the solution $x=-2$, we can plot the point on the number line that is 2 units to the left of zero. This is because $x=-2$ is 2 units away from zero on the number line.

For the solution $x=-6$, we can plot the point on the number line that is 6 units to the left of zero. This is because $x=-6$ is 6 units away from zero on the number line.

Conclusion

In this 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. We found that the solutions to the equation are $x=-2$ and $x=-6$, and we plotted these solutions on the number line. The number line representation can help us understand the relationship between the solutions and can be a useful tool for solving absolute value equations.

Final Answer

The number line that represents the solutions to the equation $|x+4|=2$ is the number line that has the points $x=-2$ and $x=-6$ marked on it.

Frequently Asked Questions

• What is an absolute value equation? An absolute value equation is an equation that involves the absolute value of a variable or expression.
• How do you solve an absolute value equation? 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.
• What is the number line representation of the solutions to the equation $|x+4|=2$? The number line representation of the solutions to the equation $|x+4|=2$ is the number line that has the points $x=-2$ and $x=-6$ marked on it.
  

Introduction

Absolute value equations are a fundamental concept in mathematics that deals with the distance of a number from zero on the number line. In this article, we will explore the concept of absolute value equations and how to solve them. We will also provide a comprehensive guide to understanding and solving absolute value equations, including a Q&A section that answers some of the most frequently asked questions.

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.

To solve an absolute value equation, 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 Absolute Value Equations

Solving absolute value equations involves considering two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative.

Case 1: Expression Inside Absolute Value is Positive

If the expression inside the absolute value is positive, then we can write the equation as $x=a$, where $a$ is a positive number. To solve for $x$, we can simply set $x$ equal to $a$.

Case 2: Expression Inside Absolute Value is Negative

If the expression inside the absolute value is negative, then we can write the equation as $x=-a$, where $a$ is a positive number. To solve for $x$, we can simply set $x$ equal to $-a$.

Examples of Solving Absolute Value Equations

Example 1: $|x-3|=4$

To solve this equation, 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.

If the expression inside the absolute value is positive, then we can write the equation as $x-3=4$. To solve for $x$, we can add 3 to both sides of the equation:

$x-3+3=4+3$

$x=7$

If the expression inside the absolute value is negative, then we can write the equation as $x-3=-4$. To solve for $x$, we can add 3 to both sides of the equation:

$x-3+3=-4+3$

$x=-1$

So, the solutions to the equation $|x-3|=4$ are $x=7$ and $x=-1$.

Example 2: $|2x+1|=3$

To solve this equation, 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.

If the expression inside the absolute value is positive, then we can write the equation as $2x+1=3$. To solve for $x$, we can subtract 1 from both sides of the equation:

$2x+1-1=3-1$

$2x=2$

To solve for $x$, we can divide both sides of the equation by 2:

$2x/2=2/2$

$x=1$

If the expression inside the absolute value is negative, then we can write the equation as $2x+1=-3$. To solve for $x$, we can subtract 1 from both sides of the equation:

$2x+1-1=-3-1$

$2x=-4$

To solve for $x$, we can divide both sides of the equation by 2:

$2x/2=-4/2$

$x=-2$

So, the solutions to the equation $|2x+1|=3$ are $x=1$ and $x=-2$.

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.

Q: How do you 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 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.

Q: How do you determine the number line representation of the solutions to an absolute value equation?

A: To determine the number line representation of the solutions to an absolute value equation, you need to plot the solutions on the number line. The number line has a scale that represents the distance of a number from zero.

Q: What is the final answer to the equation $|x+4|=2$?

A: The final answer to the equation $|x+4|=2$ is $x=-2$ and $x=-6$.

Q: How do you solve an absolute value equation with a variable inside the absolute value?

A: To solve an absolute value equation with a variable inside the absolute value, 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 an absolute value equation and a linear equation?

A: An absolute value equation is one that involves the absolute value of a variable or expression, while a linear equation is one that involves a linear expression.

Q: How do you graph an absolute value equation on a number line?

A: To graph an absolute value equation on a number line, you need to plot the solutions on the number line. The number line has a scale that represents the distance of a number from zero.

Conclusion

In this article, we have explored the concept of absolute value equations and how to solve them. We have also provided a comprehensive guide to understanding and solving absolute value equations, including a Q&A section that answers some of the most frequently asked questions. We hope that this article has been helpful in understanding and solving absolute value equations.