Solve The Inequality:$\[ -4|6b - 8| \leq 12 \\]

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Introduction

In this article, we will delve into the world of absolute value inequalities and explore the process of solving them. Absolute value inequalities involve absolute value expressions, which are expressions enclosed within double vertical bars. These inequalities can be challenging to solve, but with a clear understanding of the concepts and a step-by-step approach, we can tackle them with confidence.

Understanding Absolute Value

Before we dive into solving absolute value inequalities, 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:

−4∣6b−8∣≤12{ -4|6b - 8| \leq 12 }

Our goal is to solve for the variable b.

Step 1: Isolate the Absolute Value Expression

To solve the inequality, we need to isolate the absolute value expression. We can do this by dividing both sides of the inequality by -4. However, we need to be careful when dividing by a negative number, as it will change the direction of the inequality.

∣6b−8∣≥−3{ |6b - 8| \geq -3 }

Step 2: Remove the Absolute Value

Now that we have isolated the absolute value expression, we can remove it by considering two cases: one where the expression inside the absolute value is non-negative, and another where it is negative.

Case 1: Non-Negative Expression

When the expression inside the absolute value is non-negative, we can remove the absolute value by simply removing the double vertical bars.

6b−8≥−3{ 6b - 8 \geq -3 }

Case 2: Negative Expression

When the expression inside the absolute value is negative, we can remove the absolute value by changing the direction of the inequality and removing the double vertical bars.

6b−8≤−3{ 6b - 8 \leq -3 }

Solving the Inequalities

Now that we have removed the absolute value, we can solve the inequalities.

Case 1: Non-Negative Expression

To solve the inequality, we can add 8 to both sides and then divide by 6.

6b≥5{ 6b \geq 5 }

b≥56{ b \geq \frac{5}{6} }

Case 2: Negative Expression

To solve the inequality, we can add 8 to both sides and then divide by 6.

6b≤5{ 6b \leq 5 }

b≤56{ b \leq \frac{5}{6} }

Combining the Results

Since we considered two cases, we need to combine the results. The solution to the inequality is the intersection of the two cases.

56≤b≤56{ \frac{5}{6} \leq b \leq \frac{5}{6} }

Conclusion

In this article, we solved the absolute value inequality −4∣6b−8∣≤12-4|6b - 8| \leq 12. We isolated the absolute value expression, removed it by considering two cases, and then solved the resulting inequalities. The solution to the inequality is 56≤b≤56\frac{5}{6} \leq b \leq \frac{5}{6}.

Tips and Tricks

  • When solving absolute value inequalities, it's essential to isolate the absolute value expression first.
  • When removing the absolute value, consider two cases: one where the expression inside the absolute value is non-negative, and another where it is negative.
  • When solving the resulting inequalities, be careful with the direction of the inequality.

Common Mistakes

  • Failing to isolate the absolute value expression before removing it.
  • Not considering both cases when removing the absolute value.
  • Making errors when solving the resulting inequalities.

Real-World Applications

Absolute value inequalities have numerous real-world applications, including:

  • Modeling real-world situations where the magnitude of a quantity is important.
  • Solving problems involving distances, velocities, and accelerations.
  • Analyzing data and making predictions based on trends.

Practice Problems

Try solving the following absolute value inequalities:

  1. ∣2x+3∣≤5|2x + 3| \leq 5
  2. ∣x−2∣≥3|x - 2| \geq 3
  3. ∣3x−1∣≤2|3x - 1| \leq 2

Conclusion

Frequently Asked Questions

In this article, we will address some of the most common questions related to absolute value inequalities.

Q: What is an absolute value inequality?

A: An absolute value inequality is an inequality that involves an absolute value expression. It is an expression enclosed within double vertical bars, and it represents the distance of a number from zero on the number line.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to isolate the absolute value expression, remove it by considering two cases, and then solve the resulting inequalities.

Q: What are the two cases when removing the absolute value?

A: When removing the absolute value, you need to consider two cases: one where the expression inside the absolute value is non-negative, and another where it 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 the first case. If the expression is negative, you use the second case.

Q: What if the expression inside the absolute value is zero?

A: If the expression inside the absolute value is zero, you can remove the absolute value by simply removing the double vertical bars. However, you need to be careful when solving the resulting inequality, as it may have multiple solutions.

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

A: Yes, you can use absolute value inequalities to solve systems of equations. By using absolute value inequalities, you can find the intersection of the solution sets of the individual equations.

Q: Are absolute value inequalities used in real-world applications?

A: Yes, absolute value inequalities have numerous real-world applications, including modeling real-world situations, solving problems involving distances, velocities, and accelerations, and analyzing data and making predictions based on trends.

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 isolate the absolute value expression, not considering both cases when removing the absolute value, and making errors when solving the resulting inequalities.

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

A: Yes, you can use absolute value inequalities to solve quadratic equations. By using absolute value inequalities, you can find the solutions to the quadratic equation.

Q: Are there any online resources available to help me learn more about absolute value inequalities?

A: Yes, there are many online resources available to help you learn more about absolute value inequalities, including video tutorials, practice problems, and online courses.

Conclusion

In conclusion, absolute value inequalities are a powerful tool for solving mathematical problems. By understanding the concepts and techniques involved in solving absolute value inequalities, you can tackle even the most challenging problems. We hope this Q&A article has been helpful in addressing some of the most common questions related to absolute value inequalities.

Additional Resources

  • Khan Academy: Absolute Value Inequalities
  • Mathway: Absolute Value Inequalities
  • Wolfram Alpha: Absolute Value Inequalities
  • MIT OpenCourseWare: Absolute Value Inequalities

Practice Problems

Try solving the following absolute value inequalities:

  1. ∣2x+3∣≤5|2x + 3| \leq 5
  2. ∣x−2∣≥3|x - 2| \geq 3
  3. ∣3x−1∣≤2|3x - 1| \leq 2

Conclusion

We hope this Q&A article has been helpful in addressing some of the most common questions related to absolute value inequalities. Remember to practice solving absolute value inequalities to become more confident and proficient in solving them.