Solve The Following Inequality: ∣ 1 − 4 X ∣ − 9 \textless − 6 |1 - 4x| - 9 \ \textless \ -6 ∣1 − 4 X ∣ − 9 \textless − 6 Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice:A. There Are Finitely Many Solutions. The Solution Set Is { □ } \{\square\} { □ } .

Introduction

In this article, we will delve into the world of absolute value inequalities and focus on solving the inequality $|1 - 4x| - 9 < -6$. Absolute value inequalities can be challenging to solve, but with a clear understanding of the concepts and a step-by-step approach, we can tackle even the most complex problems.

Understanding Absolute Value

Before we dive into solving the inequality, let's take a moment to understand what absolute value means. The absolute value of a number is its distance from zero on the number line. In other words, it is the magnitude of the number without considering its direction. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

The Given Inequality

The given inequality is $|1 - 4x| - 9 < -6$. Our goal is to solve for x and find the solution set.

Step 1: Isolate the Absolute Value Expression

To solve the inequality, we need to isolate the absolute value expression. We can do this by adding 9 to both sides of the inequality:

$$|1 - 4x| < -6 + 9$$

This simplifies to:

$$|1 - 4x| < 3$$

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 one where it is negative.

Case 1: $1 - 4x \geq 0$

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

$$1 - 4x < 3$$

Subtracting 1 from both sides gives us:

$$-4x < 2$$

Dividing both sides by -4 (and flipping the inequality sign) gives us:

$$x > -\frac{1}{2}$$

However, we also need to consider the condition $1 - 4x \geq 0$. Solving this inequality gives us:

$$1 \geq 4x$$

Dividing both sides by 4 gives us:

$$\frac{1}{4} \geq x$$

Combining these two inequalities, we get:

$$\frac{1}{4} \geq x > -\frac{1}{2}$$

Case 2: $1 - 4x < 0$

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

$$-(1 - 4x) < 3$$

Simplifying this gives us:

$$-1 + 4x < 3$$

Adding 1 to both sides gives us:

$$4x < 4$$

Dividing both sides by 4 gives us:

$$x < 1$$

However, we also need to consider the condition $1 - 4x < 0$. Solving this inequality gives us:

$$1 < 4x$$

Dividing both sides by 4 gives us:

$$\frac{1}{4} < x$$

Combining these two inequalities, we get:

$$\frac{1}{4} < x < 1$$

Combining the Two Cases

Now that we have solved both cases, we need to combine the solution sets. The solution set for Case 1 is $\frac{1}{4} \geq x > -\frac{1}{2}$, and the solution set for Case 2 is $\frac{1}{4} < x < 1$.

However, we notice that the two solution sets overlap. The intersection of the two solution sets is $\frac{1}{4} < x < 1$.

Conclusion

In conclusion, the solution set for the inequality $|1 - 4x| - 9 < -6$ is $\frac{1}{4} < x < 1$. This means that the inequality is true for all values of x between $\frac{1}{4}$ and 1, exclusive.

Final Answer

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Introduction

In our previous article, we explored the concept of absolute value inequalities and solved the inequality $|1 - 4x| - 9 < -6$. In this article, we will address some common questions and concerns that students may have when dealing with absolute value inequalities.

Q&A

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

A: An absolute value equation is an equation that involves an absolute value expression, whereas an absolute value inequality is an inequality that involves an absolute value expression. In an absolute value equation, we are looking for a specific value of x that makes the absolute value expression equal to a certain value, whereas in an absolute value inequality, we are looking for a range of values of x that make the absolute value expression less than, greater than, or equal to a certain value.

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

A: When solving an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative. To determine which case to use, you need to consider the sign of the expression inside the absolute value. If the expression is positive, you use the first case. If the expression is negative, you use the second case.

Q: What if the absolute value expression is equal to a certain value?

A: If the absolute value expression is equal to a certain value, you need to consider two possibilities: one where the expression inside the absolute value is positive, and one where it is negative. In this case, you need to solve two separate equations: one where the expression inside the absolute value is positive, and one where it is negative.

Q: Can I use the same method to solve absolute value inequalities with different coefficients?

A: Yes, you can use the same method to solve absolute value inequalities with different coefficients. The key is to isolate the absolute value expression and then consider two cases: one where the expression inside the absolute value is positive, and one where it is negative.

Q: What if the absolute value inequality has a fraction or a decimal coefficient?

A: If the absolute value inequality has a fraction or a decimal coefficient, you can still use the same method to solve it. The key is to isolate the absolute value expression and then consider two cases: one where the expression inside the absolute value is positive, and one where it is negative.

Q: Can I use absolute value inequalities to solve problems in real-life situations?

A: Yes, you can use absolute value inequalities to solve problems in real-life situations. For example, you can use absolute value inequalities to model the distance between two objects, the amount of money spent on a purchase, or the number of people attending an event.

Conclusion

In conclusion, absolute value inequalities can be challenging to solve, but with a clear understanding of the concepts and a step-by-step approach, you can tackle even the most complex problems. By following the steps outlined in this article, you can solve absolute value inequalities and apply them to real-life situations.

Common Mistakes to Avoid

When solving absolute value inequalities, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not isolating the absolute value expression: Make sure to isolate the absolute value expression before considering two cases.
• Not considering both cases: Make sure to consider both cases: one where the expression inside the absolute value is positive, and one where it is negative.
• Not checking the solution set: Make sure to check the solution set to ensure that it is correct.
• Not using the correct inequality sign: Make sure to use the correct inequality sign: less than, greater than, or equal to.

Practice Problems

Here are some practice problems to help you reinforce your understanding of absolute value inequalities:

1. Solve the inequality $|2x - 3| < 5$.
2. Solve the inequality $|x + 2| > 3$.
3. Solve the inequality $|2x - 1| \geq 4$.
4. Solve the inequality $|x - 2| < 2$.
5. Solve the inequality $|x + 1| \geq 3$.

Final Answer

The final answer is: $\boxed{\left(\frac{1}{4}, 1\right)}$