Solve For All Values Of $y$ In Simplest Form.$1 = |-3y + 11|$Answer: Y = □ Y = \square Y = □

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Introduction

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Absolute value equations are a fundamental concept in algebra, and solving them requires a clear understanding of the properties of absolute value. In this article, we will focus on solving absolute value equations of the form $|ax + b| = c$, where $a$, $b$, and $c$ are constants. We will use the given equation $1 = |-3y + 11|$ as an example to demonstrate the step-by-step process of solving absolute value equations.

Understanding Absolute Value

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Absolute value is a mathematical operation that returns the distance of a number from zero on the number line. It is denoted by the symbol $|x|$, and it 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 $3$ is $3$, and the absolute value of $-3$ is also $3$.

Solving Absolute Value Equations

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

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

We will use the given equation $1 = |-3y + 11|$ as an example to demonstrate the step-by-step process of solving absolute value equations.

Case 1: $-3y + 11 = 1$

In this case, we need to isolate the variable $y$ by subtracting $11$ from both sides of the equation:

$$-3y = -10$$

Next, we need to divide both sides of the equation by $-3$ to solve for $y$:

$$y = \frac{-10}{-3}$$

Simplifying the expression, we get:

$$y = \frac{10}{3}$$

Case 2: $-3y + 11 = -1$

In this case, we need to isolate the variable $y$ by subtracting $11$ from both sides of the equation:

$$-3y = -12$$

Next, we need to divide both sides of the equation by $-3$ to solve for $y$:

$$y = \frac{-12}{-3}$$

Simplifying the expression, we get:

$$y = 4$$

Conclusion

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In this article, we have demonstrated the step-by-step process of solving absolute value equations of the form $|ax + b| = c$. We have used the given equation $1 = |-3y + 11|$ as an example to illustrate the two cases that need to be considered when solving absolute value equations. By following the steps outlined in this article, you should be able to solve absolute value equations with ease.

Frequently Asked Questions

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Q: What is the definition of absolute value?

A: Absolute value is a mathematical operation that returns the distance of a number 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: $ax + b = c$ and $ax + b = -c$.

Q: What is the difference between the two cases?

A: In Case 1, the expression inside the absolute value is equal to $c$, while in Case 2, the expression inside the absolute value is equal to $-c$.

Q: How do I isolate the variable in an absolute value equation?

A: To isolate the variable, you need to subtract the constant term from both sides of the equation and then divide both sides by the coefficient of the variable.

Example Problems

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Problem 1: Solve the absolute value equation $|2x - 3| = 5$

To solve this equation, we need to consider two cases:

• Case 1: $2x - 3 = 5$
• Case 2: $2x - 3 = -5$

Solving both cases, we get:

• Case 1: $x = 4$
• Case 2: $x = -1$

Problem 2: Solve the absolute value equation $|x + 2| = 3$

To solve this equation, we need to consider two cases:

• Case 1: $x + 2 = 3$
• Case 2: $x + 2 = -3$

Solving both cases, we get:

• Case 1: $x = 1$
• Case 2: $x = -5$

Practice Problems

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Problem 1: Solve the absolute value equation $|3y - 2| = 4$

Problem 2: Solve the absolute value equation $|2x + 1| = 6$

Problem 3: Solve the absolute value equation $|x - 1| = 2$

By practicing these problems, you should be able to become proficient in solving absolute value equations.

Final Thoughts

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Solving absolute value equations requires a clear understanding of the properties of absolute value and the ability to isolate the variable in an equation. By following the steps outlined in this article, you should be able to solve absolute value equations with ease. Remember to consider both cases when solving an absolute value equation, and don't hesitate to practice with example problems to reinforce your understanding.

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Introduction

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Absolute value equations are a fundamental concept in algebra, and solving them requires a clear understanding of the properties of absolute value. In this article, we will provide a comprehensive Q&A guide to help you understand and solve absolute value equations.

Q&A

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Q: What is the definition of absolute value?

A: Absolute value is a mathematical operation that returns the distance of a number 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: $ax + b = c$ and $ax + b = -c$.

Q: What is the difference between the two cases?

A: In Case 1, the expression inside the absolute value is equal to $c$, while in Case 2, the expression inside the absolute value is equal to $-c$.

Q: How do I isolate the variable in an absolute value equation?

A: To isolate the variable, you need to subtract the constant term from both sides of the equation and then divide both sides by the coefficient of the variable.

Q: What is the formula for solving absolute value equations?

A: The formula for solving absolute value equations is:

$$|ax + b| = c \Rightarrow ax + b = c \text{ or } ax + b = -c$$

Q: How do I determine which case to use?

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 Case 1. If the expression is negative, use Case 2.

Q: Can I use absolute value equations to solve systems of equations?

A: Yes, you can use absolute value equations to solve systems of equations. However, you need to be careful when solving the system, as the absolute value equation may have multiple solutions.

Q: How do I graph absolute value equations?

A: To graph an absolute value equation, you need to plot the two cases on a coordinate plane. The graph of the absolute value equation will be a V-shaped graph with the vertex at the point where the two cases intersect.

Q: Can I use absolute value equations to solve inequalities?

A: Yes, you can use absolute value equations to solve inequalities. However, you need to be careful when solving the inequality, as the absolute value equation may have multiple solutions.

Advanced Topics

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Q: How do I solve absolute value equations with multiple variables?

A: To solve absolute value equations with multiple variables, you need to use the same steps as solving absolute value equations with one variable. However, you need to be careful when isolating the variables, as the equation may have multiple solutions.

Q: How do I solve absolute value equations with fractions?

A: To solve absolute value equations with fractions, you need to use the same steps as solving absolute value equations with integers. However, you need to be careful when simplifying the fractions, as the equation may have multiple solutions.

Q: How do I solve absolute value equations with decimals?

A: To solve absolute value equations with decimals, you need to use the same steps as solving absolute value equations with integers. However, you need to be careful when rounding the decimals, as the equation may have multiple solutions.

Practice Problems

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Problem 1: Solve the absolute value equation $|2x - 3| = 5$

Problem 2: Solve the absolute value equation $|x + 2| = 3$

Problem 3: Solve the absolute value equation $|3y - 2| = 4$

By practicing these problems, you should be able to become proficient in solving absolute value equations.

Final Thoughts

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Solving absolute value equations requires a clear understanding of the properties of absolute value and the ability to isolate the variable in an equation. By following the steps outlined in this article, you should be able to solve absolute value equations with ease. Remember to consider both cases when solving an absolute value equation, and don't hesitate to practice with example problems to reinforce your understanding.

Resources

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

By using these resources, you should be able to gain a deeper understanding of absolute value equations and improve your problem-solving skills.