Identify Which Values Of { X $}$ Are Solutions For The Inequality { |3x - 2| \ \textless \ 4$} . . . [ \begin{array}{|l|c|} \hline \text{Value} & \text{Solution} \ \hline 0 & \square \ \hline \frac{1}{3} & \square \ \hline 1

Introduction

Absolute value inequalities are a fundamental concept in mathematics, and solving them requires a clear understanding of the properties of absolute value functions. In this article, we will focus on solving the inequality $|3x - 2| < 4$, where we need to find the values of $x$ that satisfy this inequality. We will break down the solution process into manageable steps, making it easier to understand and apply.

Understanding Absolute Value Functions

Before we dive into solving the inequality, let's briefly review the properties of absolute value functions. The absolute value function $|x|$ is defined as:

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

This function returns the distance of $x$ from 0 on the number line. For example, $|3| = 3$ and $|-3| = 3$.

Solving the Inequality

Now that we have a basic understanding of absolute value functions, let's solve the inequality $|3x - 2| < 4$. To do this, we need to consider two cases:

Case 1: $3x - 2 \geq 0$

In this case, the absolute value function can be rewritten as:

$$|3x - 2| = 3x - 2$$

Substituting this into the inequality, we get:

$$3x - 2 < 4$$

Solving for $x$, we get:

$$3x < 6$$

$$x < 2$$

However, we need to remember that this solution is only valid when $3x - 2 \geq 0$. To find the values of $x$ that satisfy this condition, we need to solve the inequality:

$$3x - 2 \geq 0$$

Solving for $x$, we get:

$$3x \geq 2$$

$$x \geq \frac{2}{3}$$

Therefore, the solution to this case is:

$$\frac{2}{3} \leq x < 2$$

Case 2: $3x - 2 < 0$

In this case, the absolute value function can be rewritten as:

$$|3x - 2| = -(3x - 2)$$

Substituting this into the inequality, we get:

$$-(3x - 2) < 4$$

Simplifying, we get:

$$-3x + 2 < 4$$

$$-3x < 2$$

$$x > -\frac{2}{3}$$

However, we need to remember that this solution is only valid when $3x - 2 < 0$. To find the values of $x$ that satisfy this condition, we need to solve the inequality:

$$3x - 2 < 0$$

Solving for $x$, we get:

$$3x < 2$$

$$x < \frac{2}{3}$$

Therefore, the solution to this case is:

$$x < \frac{2}{3}$$

Combining the Solutions

Now that we have solved the inequality for both cases, we can combine the solutions to find the final answer. The solution to the inequality $|3x - 2| < 4$ is:

$$\frac{2}{3} < x < 2$$

Conclusion

Solving absolute value inequalities requires a clear understanding of the properties of absolute value functions. By breaking down the solution process into manageable steps, we can make it easier to understand and apply. In this article, we solved the inequality $|3x - 2| < 4$ and found the values of $x$ that satisfy this inequality. We hope that this article has provided a clear and concise guide to solving absolute value inequalities.

Discussion

• What are some common mistakes to avoid when solving absolute value inequalities?
• How can we use absolute value inequalities in real-world applications?
• What are some tips for simplifying absolute value expressions?

Answer Key

• The solution to the inequality $|3x - 2| < 4$ is $\frac{2}{3} < x < 2$.
• Some common mistakes to avoid when solving absolute value inequalities include:
  • Failing to consider both cases when solving the inequality.
  • Not simplifying the absolute value expression before solving the inequality.
  • Not checking the validity of the solution.
• Absolute value inequalities can be used in real-world applications such as:
  • Modeling the distance between two points on a number line.
  • Finding the range of values for a variable that satisfies a certain condition.
  • Solving problems involving motion or velocity.

Additional Resources

• Khan Academy: Absolute Value Inequalities
• Mathway: Absolute Value Inequality Solver
• Wolfram Alpha: Absolute Value Inequality Calculator
  Absolute Value Inequality Q&A =============================

Q: What is an absolute value inequality?

A: An absolute value inequality is an inequality that involves the absolute value of a variable or expression. It is a mathematical statement that compares the absolute value of an expression to a certain value or expression.

Q: How do I solve an absolute value inequality?

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

1. The expression inside the absolute value is non-negative (i.e., greater than or equal to 0).
2. The expression inside the absolute value is negative (i.e., less than 0).

For each case, you need to solve the inequality separately and then combine the solutions.

Q: What are some common mistakes to avoid when solving absolute value inequalities?

A: Some common mistakes to avoid when solving absolute value inequalities include:

• Failing to consider both cases when solving the inequality.
• Not simplifying the absolute value expression before solving the inequality.
• Not checking the validity of the solution.

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

A: To determine which case to use, you need to consider the sign of the expression inside the absolute value. If the expression is non-negative, use the first case. If the expression is negative, use the second case.

Q: Can I use absolute value inequalities to solve problems involving motion or velocity?

A: Yes, absolute value inequalities can be used to solve problems involving motion or velocity. For example, you can use absolute value inequalities to model the distance between two points on a number line or to find the range of values for a variable that satisfies a certain condition.

Q: How do I use absolute value inequalities in real-world applications?

A: Absolute value inequalities can be used in a variety of real-world applications, including:

• Modeling the distance between two points on a number line.
• Finding the range of values for a variable that satisfies a certain condition.
• Solving problems involving motion or velocity.

Q: What are some tips for simplifying absolute value expressions?

A: Some tips for simplifying absolute value expressions include:

• Using the definition of absolute value to rewrite the expression.
• Simplifying the expression inside the absolute value.
• Using algebraic manipulations to simplify the expression.

Q: Can I use absolute value inequalities to solve problems involving finance or economics?

A: Yes, absolute value inequalities can be used to solve problems involving finance or economics. For example, you can use absolute value inequalities to model the value of a stock or to find the range of values for a variable that satisfies a certain condition.

Q: How do I check the validity of a solution to an absolute value inequality?

A: To check the validity of a solution to an absolute value inequality, you need to make sure that the solution satisfies the original inequality. You can do this by plugging the solution back into the original inequality and checking if it is true.

Q: What are some common applications of absolute value inequalities in science and engineering?

A: Some common applications of absolute value inequalities in science and engineering include:

• Modeling the distance between two points in space.
• Finding the range of values for a variable that satisfies a certain condition.
• Solving problems involving motion or velocity.

Q: Can I use absolute value inequalities to solve problems involving data analysis or statistics?

A: Yes, absolute value inequalities can be used to solve problems involving data analysis or statistics. For example, you can use absolute value inequalities to model the distribution of a dataset or to find the range of values for a variable that satisfies a certain condition.

Q: How do I use absolute value inequalities to solve problems involving optimization or minimization?

A: Absolute value inequalities can be used to solve problems involving optimization or minimization by modeling the objective function as an absolute value expression and then solving the resulting inequality.