Solve The Inequality: ∣ 4 U − 2 ∣ − 6 \textgreater 8 |4u - 2| - 6 \ \textgreater \ 8 ∣4 U − 2∣ − 6 \textgreater 8 A. W \textgreater 4 W \ \textgreater \ 4 W \textgreater 4 Or W \textless − 3 W \ \textless \ -3 W \textless − 3 B. − 3 \textless M \textless 4 -3 \ \textless \ M \ \textless \ 4 − 3 \textless M \textless 4 C. W \textgreater 4 W \ \textgreater \ 4 W \textgreater 4 D. $w \ \textgreater \

Introduction

Inequalities are mathematical expressions that compare two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we will focus on solving the inequality $|4u - 2| - 6 \ \textgreater \ 8$.

Understanding Absolute Value

Before we dive into solving the inequality, 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, without considering direction. In other words, the absolute value of a number is always non-negative. The absolute value of a number $x$ is denoted by $|x|$.

Step 1: Isolate the Absolute Value Expression

To solve the inequality $|4u - 2| - 6 \ \textgreater \ 8$, we need to isolate the absolute value expression. We can do this by adding 6 to both sides of the inequality:

$$|4u - 2| \ \textgreater \ 14$$

Step 2: Split the Inequality into Two Cases

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

Case 1: $4u - 2 \ \textgreater \ 0$

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

$$4u - 2 \ \textgreater \ 14$$

We can solve this inequality by adding 2 to both sides:

$$4u \ \textgreater \ 16$$

Dividing both sides by 4, we get:

$$u \ \textgreater \ 4$$

Case 2: $4u - 2 \ \textless \ 0$

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

$$-(4u - 2) \ \textgreater \ 14$$

Simplifying the expression, we get:

$$-4u + 2 \ \textgreater \ 14$$

Subtracting 2 from both sides, we get:

$$-4u \ \textgreater \ 12$$

Dividing both sides by -4, we need to reverse the direction of the inequality:

$$u \ \textless \ -3$$

Combining the Two Cases

We have two cases: $u \ \textgreater \ 4$ and $u \ \textless \ -3$. Since these two cases are mutually exclusive, we can combine them using the "or" operator:

$$u \ \textgreater \ 4 \ \text{or} \ u \ \textless \ -3$$

Conclusion

Solving the inequality $|4u - 2| - 6 \ \textgreater \ 8$ involves isolating the absolute value expression, splitting the inequality into two cases, and combining the results. The final solution is $u \ \textgreater \ 4 \ \text{or} \ u \ \textless \ -3$.

Answer

The correct answer is:

A. $w \ \textgreater \ 4$ or $w \ \textless \ -3$

Discussion

This problem requires a good understanding of absolute value and how to solve inequalities. The key concept is to isolate the absolute value expression and then split the inequality into two cases. By combining the two cases, we can find the final solution.

Tips and Variations

• When dealing with absolute value inequalities, it's essential to consider both cases: one where the expression inside the absolute value is positive, and another where it is negative.
• When combining the two cases, use the "or" operator to ensure that the final solution is correct.
• Practice solving absolute value inequalities to become more comfortable with the concept.

Common Mistakes

• Failing to isolate the absolute value expression.
• Not considering both cases when dealing with absolute value inequalities.
• Not combining the two cases correctly.

Conclusion

Q&A: Solving Inequalities with Absolute Value

Q: What is the first step in solving an inequality with absolute value? A: The first step in solving an inequality with absolute value is to isolate the absolute value expression. This involves moving any constants or variables that are not part of the absolute value expression to the other side of the inequality.

Q: How do I split the inequality into two cases? A: To split the inequality into two cases, you need to consider two scenarios: one where the expression inside the absolute value is positive, and another where it is negative. This will give you two separate inequalities to solve.

Q: What is the difference between the two cases? A: The two cases are based on the sign of the expression inside the absolute value. If the expression is positive, the absolute value expression is equal to the expression itself. If the expression is negative, the absolute value expression is equal to the negative of the expression.

Q: How do I combine the two cases? A: To combine the two cases, you need to use the "or" operator. This means that the final solution will be a combination of the two separate inequalities.

Q: What are some common mistakes to avoid when solving inequalities with absolute value? A: Some common mistakes to avoid when solving inequalities with absolute value include:

• Failing to isolate the absolute value expression
• Not considering both cases when dealing with absolute value inequalities
• Not combining the two cases correctly

Q: Can you give an example of how to solve an inequality with absolute value? A: Let's consider the inequality $|x - 3| \ \textgreater \ 5$. To solve this inequality, we need to isolate the absolute value expression, split the inequality into two cases, and combine the results.

Step 1: Isolate the Absolute Value Expression

$$|x - 3| \ \textgreater \ 5$$

Step 2: Split the Inequality into Two Cases

Case 1: $x - 3 \ \textgreater \ 0$

$$x - 3 \ \textgreater \ 5$$

Case 2: $x - 3 \ \textless \ 0$

$$-(x - 3) \ \textgreater \ 5$$

Step 3: Solve the Two Cases

Case 1: $x - 3 \ \textgreater \ 0$

$$x \ \textgreater \ 8$$

Case 2: $x - 3 \ \textless \ 0$

$$x \ \textless \ -2$$

Step 4: Combine the Two Cases

$$x \ \textgreater \ 8 \ \text{or} \ x \ \textless \ -2$$

Q: What is the final answer to the inequality $|x - 3| \ \textgreater \ 5$? A: The final answer to the inequality $|x - 3| \ \textgreater \ 5$ is $x \ \textgreater \ 8 \ \text{or} \ x \ \textless \ -2$.

Conclusion

Solving inequalities with absolute value requires a good understanding of the concept and the ability to isolate the absolute value expression, split the inequality into two cases, and combine the results. By following the steps outlined in this article, you can become more confident in solving absolute value inequalities.