(a) The Solution Set Of $|x-6|=1$ Is $\square$.

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Introduction

In mathematics, absolute value equations are a type of algebraic equation that involves the absolute value of a variable or expression. These equations are often used to model real-world problems, such as distance, temperature, and financial applications. In this article, we will focus on solving absolute value equations, specifically the equation $|x-6|=1$. We will explore the concept of absolute value, learn how to solve absolute value equations, and find the solution set of the given equation.

Understanding Absolute Value

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line. It is denoted by the symbol $|x|$ and is defined as:

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

For example, the absolute value of 5 is 5, and the absolute value of -3 is 3.

Solving Absolute Value Equations

To solve an absolute value equation, we need to consider two cases: one where the expression inside the absolute value is non-negative, and another where the expression is negative.

Let's consider the equation $|x-6|=1$. We can start by setting up two equations:

$$x-6=1 \quad \text{or} \quad x-6=-1$$

Solving the first equation, we get:

$$x-6=1 \Rightarrow x=7$$

Solving the second equation, we get:

$$x-6=-1 \Rightarrow x=5$$

Therefore, the solution set of the equation $|x-6|=1$ is $\boxed{\{5, 7\}}$.

Graphical Representation

To visualize the solution set, we can graph the equation $|x-6|=1$ on a number line. The graph will consist of two points: 5 and 7.

Conclusion

In this article, we learned how to solve absolute value equations, specifically the equation $|x-6|=1$. We explored the concept of absolute value, learned how to set up and solve absolute value equations, and found the solution set of the given equation. The solution set of the equation $|x-6|=1$ is $\boxed{\{5, 7\}}$.

Applications of Absolute Value Equations

Absolute value equations have numerous applications in real-world problems, such as:

• Distance: The distance between two points on a number line can be represented by an absolute value equation.
• Temperature: The temperature of a system can be represented by an absolute value equation.
• Financial applications: The value of a stock or a currency can be represented by an absolute value equation.

Tips and Tricks

When solving absolute value equations, it's essential to remember the following tips and tricks:

• Always set up two equations: one where the expression inside the absolute value is non-negative, and another where the expression is negative.
• Solve each equation separately.
• Check your solutions by plugging them back into the original equation.

Practice Problems

Here are some practice problems to help you reinforce your understanding of absolute value equations:

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

Conclusion

In conclusion, absolute value equations are a fundamental concept in mathematics that has numerous applications in real-world problems. By understanding how to solve absolute value equations, we can model and solve a wide range of problems. In this article, we learned how to solve the equation $|x-6|=1$ and found the solution set of the given equation. We also explored the concept of absolute value, learned how to set up and solve absolute value equations, and discussed the applications and tips and tricks for solving absolute value equations.

Introduction

In the previous article, we explored the concept of absolute value equations, specifically the equation $|x-6|=1$. We learned how to solve absolute value equations, found the solution set of the given equation, and discussed the applications and tips and tricks for solving absolute value equations. In this article, we will answer some frequently asked questions (FAQs) about absolute value equations.

Q&A

Q1: What is the definition of absolute value?

A1: The absolute value of a number is its distance from zero on the number line. It is denoted by the symbol $|x|$ and is defined as:

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

Q2: How do I solve an absolute value equation?

A2: To solve an absolute value equation, you need to consider two cases: one where the expression inside the absolute value is non-negative, and another where the expression is negative. You can set up two equations:

$$x-a=b \quad \text{or} \quad x-a=-b$$

where $a$ and $b$ are constants.

Q3: What is the solution set of the equation $|x-6|=1$?

A3: The solution set of the equation $|x-6|=1$ is $\boxed{\{5, 7\}}$.

Q4: How do I graph an absolute value equation?

A4: To graph an absolute value equation, you can use a number line. The graph will consist of two points: the solution to the first equation and the solution to the second equation.

Q5: What are some applications of absolute value equations?

A5: Absolute value equations have numerous applications in real-world problems, such as:

• Distance: The distance between two points on a number line can be represented by an absolute value equation.
• Temperature: The temperature of a system can be represented by an absolute value equation.
• Financial applications: The value of a stock or a currency can be represented by an absolute value equation.

Q6: What are some tips and tricks for solving absolute value equations?

A6: Here are some tips and tricks for solving absolute value equations:

• Always set up two equations: one where the expression inside the absolute value is non-negative, and another where the expression is negative.
• Solve each equation separately.
• Check your solutions by plugging them back into the original equation.

Q7: How do I solve an absolute value inequality?

A7: To solve an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value is non-negative, and another where the expression is negative. You can set up two inequalities:

$$x-a\geq b \quad \text{or} \quad x-a\leq -b$$

where $a$ and $b$ are constants.

Q8: What is the difference between an absolute value equation and an absolute value inequality?

A8: An absolute value equation is an equation that involves an absolute value expression, while an absolute value inequality is an inequality that involves an absolute value expression.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about absolute value equations. We discussed the definition of absolute value, how to solve absolute value equations, the solution set of the equation $|x-6|=1$, how to graph an absolute value equation, some applications of absolute value equations, and some tips and tricks for solving absolute value equations. We also discussed how to solve absolute value inequalities and the difference between an absolute value equation and an absolute value inequality.

Practice Problems

Here are some practice problems to help you reinforce your understanding of absolute value equations and inequalities:

• Solve the equation $|x+3|=2$.
• Solve the inequality $|x-2|\geq 4$.
• Solve the equation $|x-5|=1$.
• Solve the inequality $|x+1|\leq 3$.

Resources

Here are some resources to help you learn more about absolute value equations and inequalities:

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