Select The Correct Answer.What Is The Solution To The Equation $|5-x|=13$?A. $x=8$ B. $x=-18$ C. $x=-18$ Or $x=8$ D. $x=-8$ Or $x=18$

Introduction

Absolute value equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the properties of absolute value. In this article, we will focus on solving the equation $|5-x|=13$, which is a classic example of an absolute value equation. We will break down the solution step by step, and by the end of this article, you will be able to solve similar equations with ease.

Understanding Absolute Value

Before we dive into the solution, let's quickly review the concept of absolute value. The absolute value of a number $a$, denoted by $|a|$, is the distance of $a$ from zero on the number line. In other words, it is the magnitude of $a$ without considering its direction. For example, the absolute value of $-3$ is $3$, and the absolute value of $4$ is also $4$.

The Equation $|5-x|=13$

Now that we have a good understanding of absolute value, let's tackle the equation $|5-x|=13$. To solve this equation, we need to consider two cases: when $5-x$ is positive and when $5-x$ is negative.

Case 1: $5-x$ is Positive

When $5-x$ is positive, we can simply remove the absolute value sign and solve the equation as follows:

$$5-x=13$$

To solve for $x$, we need to isolate $x$ on one side of the equation. We can do this by subtracting $5$ from both sides of the equation:

$$-x=13-5$$

$$-x=8$$

Now, we can multiply both sides of the equation by $-1$ to solve for $x$:

$$x=-8$$

So, in this case, the solution to the equation is $x=-8$.

Case 2: $5-x$ is Negative

When $5-x$ is negative, we need to multiply the absolute value sign by $-1$ to make it positive. This gives us the following equation:

$$-(5-x)=13$$

To simplify this equation, we can distribute the negative sign to the terms inside the parentheses:

$$-5+x=13$$

Now, we can add $5$ to both sides of the equation to isolate $x$:

$$x=13+5$$

$$x=18$$

So, in this case, the solution to the equation is $x=18$.

Conclusion

In conclusion, the solution to the equation $|5-x|=13$ is $x=-8$ or $x=18$. We arrived at this solution by considering two cases: when $5-x$ is positive and when $5-x$ is negative. By following the steps outlined in this article, you should be able to solve similar absolute value equations with ease.

Practice Problems

To reinforce your understanding of absolute value equations, try solving the following practice problems:

1. Solve the equation $|2x-3|=5$.
2. Solve the equation $|x+2|=3$.
3. Solve the equation $|3x-2|=4$.

Answer Key

1. $x=4$ or $x=-2$
2. $x=1$ or $x=-5$
3. $x=2$ or $x=-\frac{2}{3}$

Final Thoughts

Introduction

In our previous article, we explored the concept of absolute value equations and solved the equation $|5-x|=13$. In this article, we will continue to delve deeper into the world of absolute value equations and answer some of the most frequently asked questions.

Q&A

Q: What is the definition of absolute value?

A: The absolute value of a number $a$, denoted by $|a|$, is the distance of $a$ from zero on the number line. In other words, it is the magnitude of $a$ without considering its direction.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases: when the expression inside the absolute value sign is positive and when it is negative. You can then use algebraic techniques to solve for the variable.

Q: What is the difference between an absolute value equation and a linear equation?

A: An absolute value equation is an equation that contains an absolute value expression, whereas a linear equation is an equation that can be written in the form $ax+b=c$, where $a$, $b$, and $c$ are constants.

Q: Can I use the same techniques to solve absolute value inequalities as I do to solve absolute value equations?

A: No, the techniques used to solve absolute value inequalities are different from those used to solve absolute value equations. Absolute value inequalities involve finding the values of the variable that make the inequality true, whereas absolute value equations involve finding the values of the variable that make the equation true.

Q: How do I know which case to use when solving an absolute value equation?

A: When solving an absolute value equation, you need to consider two cases: when the expression inside the absolute value sign is positive and when it is negative. You can use the following rule to determine which case to use:

• If the expression inside the absolute value sign is positive, use the case where the absolute value sign is removed.
• If the expression inside the absolute value sign is negative, use the case where the absolute value sign is multiplied by $-1$.

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

A: Yes, absolute value equations can be used to model real-world problems that involve distances, temperatures, and other quantities that can be represented as absolute values.

Q: How do I graph absolute value equations?

A: To graph an absolute value equation, you can use the following steps:

• Plot the points on the graph that correspond to the solutions of the equation.
• Draw a line through the points to form the graph of the equation.

Practice Problems

To reinforce your understanding of absolute value equations, try solving the following practice problems:

1. Solve the equation $|x-2|=3$.
2. Solve the equation $|2x+1|=4$.
3. Solve the equation $|x-1|=2$.

Answer Key

1. $x=5$ or $x=-1$
2. $x=1.5$ or $x=-2.5$
3. $x=3$ or $x=-1$

Final Thoughts

Absolute value equations are a powerful tool for modeling real-world problems and solving equations. By understanding the properties of absolute value and using the techniques outlined in this article, you can solve absolute value equations with ease. Remember to consider two cases: when the expression inside the absolute value sign is positive and when it is negative. With practice, you will become more comfortable solving absolute value equations and be able to tackle more complex problems with confidence.