Select The Correct Answer.Solve: ∣ 11 X + 10 ∣ = 22 |11x + 10| = 22 ∣11 X + 10∣ = 22 A. X = − 12 11 X = -\frac{12}{11} X = − 11 12 ​ Or X = 32 11 X = \frac{32}{11} X = 11 32 ​ B. X = − 10 11 X = -\frac{10}{11} X = − 11 10 ​ Or X = 10 11 X = \frac{10}{11} X = 11 10 ​ C. X = − 11 32 X = -\frac{11}{32} X = − 32 11 ​ Or $x =

Introduction

In mathematics, absolute value equations are a type of equation that involves the absolute value of a variable or expression. These equations can be solved using various techniques, including algebraic manipulation and graphical methods. In this article, we will focus on solving absolute value equations of the form $|ax + b| = c$, where $a$, $b$, and $c$ are constants.

Understanding Absolute Value

Before we dive into solving absolute value equations, it's essential to understand the concept of absolute value. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of $-3$ is $3$, and the absolute value of $5$ is also $5$. In mathematical notation, the absolute value of a number $x$ is denoted by $|x|$.

Solving Absolute Value Equations

To solve an absolute value equation of the form $|ax + b| = c$, we need to consider two cases:

Case 1: $ax + b = c$

In this case, we can simply solve for $x$ by isolating it on one side of the equation.

Case 2: $ax + b = -c$

In this case, we can also solve for $x$ by isolating it on one side of the equation.

Solving the Given Equation

Now, let's apply the above techniques to solve the given equation: $|11x + 10| = 22$.

Case 1: $11x + 10 = 22$

To solve for $x$, we need to isolate it on one side of the equation. We can do this by subtracting $10$ from both sides of the equation:

$$11x + 10 - 10 = 22 - 10$$

This simplifies to:

$$11x = 12$$

Next, we can divide both sides of the equation by $11$ to solve for $x$:

$$\frac{11x}{11} = \frac{12}{11}$$

This gives us:

$$x = \frac{12}{11}$$

Case 2: $11x + 10 = -22$

To solve for $x$, we need to isolate it on one side of the equation. We can do this by subtracting $10$ from both sides of the equation:

$$11x + 10 - 10 = -22 - 10$$

This simplifies to:

$$11x = -32$$

Next, we can divide both sides of the equation by $11$ to solve for $x$:

$$\frac{11x}{11} = \frac{-32}{11}$$

This gives us:

$$x = -\frac{32}{11}$$

Conclusion

In conclusion, we have solved the given absolute value equation $|11x + 10| = 22$ using the two-case technique. We found that the solutions are $x = \frac{12}{11}$ and $x = -\frac{32}{11}$.

Final Answer

The final answer is:

A. $x = -\frac{12}{11}$ or $x = \frac{32}{11}$

This matches option A in the given problem statement.

Discussion

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Introduction

In our previous article, we discussed how to solve absolute value equations of the form $|ax + b| = c$. In this article, we will answer some frequently asked questions about absolute value equations.

Q: What is the definition of an absolute value equation?

A: An absolute value equation is an equation that involves the absolute value of a variable or expression. It is a type of equation that can be solved using various techniques, including algebraic manipulation and graphical methods.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases:

• Case 1: $ax + b = c$
• Case 2: $ax + b = -c$

You can then solve for $x$ in each case by isolating it on one side of the equation.

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

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

A: Yes, you can use the same techniques to solve absolute value inequalities. However, you need to consider the direction of the inequality sign and the properties of absolute value.

Q: How do I graph an absolute value function?

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

1. Plot the point $(0, |b|)$ on the graph.
2. Plot the point $(0, -|b|)$ on the graph.
3. Draw a V-shaped graph that passes through the two points.

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

A: Yes, you can use absolute value equations to model real-world problems. For example, you can use absolute value equations to model the distance between two points, the cost of a product, or the temperature of a system.

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 cases when solving an absolute value equation.
• Not isolating the variable on one side of the equation.
• Not checking the solutions for extraneous solutions.

Conclusion

In conclusion, we have answered some frequently asked questions about absolute value equations. We hope that this article has provided you with a better understanding of absolute value equations and how to solve them.

Final Tips

• Make sure to consider both cases when solving an absolute value equation.
• Isolate the variable on one side of the equation.
• Check the solutions for extraneous solutions.

Common Absolute Value Equations

Here are some common absolute value equations that you may encounter:

• $|x| = 5$
• $|x - 2| = 3$
• $|2x + 1| = 4$

Solving Absolute Value Equations: Practice Problems

Here are some practice problems to help you practice solving absolute value equations:

• $|x + 2| = 5$
• $|2x - 3| = 2$
• $|x - 1| = 4$

We hope that this article has provided you with a better understanding of absolute value equations and how to solve them.