What Is The Solution To The Inequality $|2x - 3| \ \textgreater \ 5$?A. $x \ \textless \ -1$ Or $x \ \textgreater \ 4$ B. $x \ \textless \ 0$ Or $x \ \textgreater \ 8$ C. $0 \ \textless \ X

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Introduction

Inequalities are mathematical expressions that compare two values or expressions, often involving variables. Solving inequalities requires a different approach than solving equations, as the goal is to find the values of the variable that satisfy the given condition. In this article, we will focus on solving the inequality $|2x - 3| \ \textgreater \ 5$, which involves absolute value. We will break down the solution step by step and provide a clear explanation of the process.

Understanding Absolute Value Inequalities

Absolute value inequalities involve the absolute value of an expression, which is always non-negative. The absolute value of a number $x$ is denoted by $|x|$ and is defined as:

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

When dealing with absolute value inequalities, we need to consider two cases: one where the expression inside the absolute value is non-negative, and another where it is negative.

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

In this case, the absolute value of $2x - 3$ is equal to $2x - 3$ itself. We can rewrite the inequality as:

$$2x - 3 > 5$$

Solving for $x$, we get:

$$2x > 8$$

$$x > 4$$

Case 2: $2x - 3 < 0$

In this case, the absolute value of $2x - 3$ is equal to $-(2x - 3)$. We can rewrite the inequality as:

$$-(2x - 3) > 5$$

Simplifying, we get:

$$-2x + 3 > 5$$

$$-2x > 2$$

$$x < -1$$

Combining the Cases

We have found the solutions to the inequality in two cases: $x > 4$ and $x < -1$. However, we need to consider the original inequality, which involves absolute value. The absolute value of an expression is always non-negative, so we need to combine the two cases in a way that takes into account the absolute value.

Solution

The solution to the inequality $|2x - 3| \ \textgreater \ 5$ is:

$$x < -1 \text{ or } x > 4$$

This means that the values of $x$ that satisfy the inequality are all real numbers less than $-1$ and all real numbers greater than $4$.

Conclusion

Solving absolute value inequalities requires considering two cases: one where the expression inside the absolute value is non-negative, and another where it is negative. By breaking down the solution step by step and providing a clear explanation of the process, we have found the solution to the inequality $|2x - 3| \ \textgreater \ 5$. The solution is $x < -1 \text{ or } x > 4$, which means that the values of $x$ that satisfy the inequality are all real numbers less than $-1$ and all real numbers greater than $4$.

Frequently Asked Questions

• Q: What is the solution to the inequality $|2x - 3| \ \textgreater \ 5$? A: The solution is $x < -1 \text{ or } x > 4$.
• Q: How do I solve absolute value inequalities? A: To solve absolute value inequalities, you need to consider two cases: one where the expression inside the absolute value is non-negative, and another where it is negative.
• Q: What is the definition of absolute value? A: The absolute value of a number $x$ is denoted by $|x|$ and is defined as:

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

Final Answer

The final answer is $x < -1 \text{ or } x > 4$.

Introduction

Solving absolute value inequalities can be a challenging task, but with the right approach, it can be made easier. In this article, we will provide a comprehensive guide to solving absolute value inequalities, including a step-by-step solution to the inequality $|2x - 3| \ \textgreater \ 5$. We will also answer some frequently asked questions about solving absolute value inequalities.

Q&A: Solving Absolute Value Inequalities

Q: What is the definition of absolute value?

A: The absolute value of a number $x$ is denoted by $|x|$ and is defined as:

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

Q: How do I solve absolute value inequalities?

A: To solve absolute value inequalities, you need to consider two cases: one where the expression inside the absolute value is non-negative, and another where it is negative. You can then use the properties of absolute value to simplify the inequality and find the solution.

Q: What are the two cases for solving absolute value inequalities?

A: The two cases for solving absolute value inequalities are:

• Case 1: The expression inside the absolute value is non-negative.
• Case 2: The expression inside the absolute value is negative.

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 non-negative, you use Case 1. If the expression is negative, you use Case 2.

Q: What is the solution to the inequality $|2x - 3| \ \textgreater \ 5$?

A: The solution to the inequality $|2x - 3| \ \textgreater \ 5$ is:

$$x < -1 \text{ or } x > 4$$

Q: How do I simplify absolute value inequalities?

A: To simplify absolute value inequalities, you can use the properties of absolute value, such as:

• $|x| = x$ if $x \geq 0$
• $|x| = -x$ if $x < 0$

You can also use algebraic manipulations, such as adding or subtracting the same value to both sides of the inequality.

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 (non-negative and negative)
• Not using the properties of absolute value correctly
• Not simplifying the inequality correctly

Q: How do I check my solution to an absolute value inequality?

A: To check your solution to an absolute value inequality, you can plug in values of $x$ that satisfy the inequality and verify that the absolute value expression is indeed greater than the given value.

Conclusion

Solving absolute value inequalities requires a clear understanding of the properties of absolute value and the ability to simplify inequalities correctly. By following the steps outlined in this article and avoiding common mistakes, you can solve absolute value inequalities with confidence.

Final Answer

The final answer is $x < -1 \text{ or } x > 4$.