Solve The Inequality: ∣ − 6 X + 2 ∣ \textless 8 |-6x + 2| \ \textless \ 8 ∣ − 6 X + 2∣ \textless 8

by ADMIN 101 views

Introduction

In mathematics, absolute value inequalities are a type of inequality that involves the absolute value of an expression. These inequalities can be challenging to solve, but with a clear understanding of the concept and a step-by-step approach, they can be tackled with ease. In this article, we will focus on solving the inequality ∣−6x+2∣<8|-6x + 2| < 8. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding Absolute Value Inequalities

Before we dive into solving the inequality, let's take a moment to understand what absolute value inequalities are. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5. When we have an absolute value inequality, we are looking for values of the variable that make the absolute value of the expression less than, greater than, or equal to a certain value.

The Given Inequality

The given inequality is ∣−6x+2∣<8|-6x + 2| < 8. This means that we are looking for values of xx that make the absolute value of −6x+2-6x + 2 less than 8.

Step 1: Write the Inequality as a Double Inequality

To solve the inequality, we can start by writing it as a double inequality. A double inequality is an inequality that has two parts: one part that is less than or equal to a certain value, and another part that is greater than or equal to the same value. In this case, we can write the inequality as:

−8<−6x+2<8-8 < -6x + 2 < 8

Step 2: Subtract 2 from Each Part of the Inequality

To isolate the term with the variable, we can subtract 2 from each part of the inequality. This gives us:

−10<−6x<6-10 < -6x < 6

Step 3: Divide Each Part of the Inequality by -6

To solve for xx, we can divide each part of the inequality by -6. However, when we divide by a negative number, we need to reverse the direction of the inequality signs. This gives us:

6−6>x>−6−6\frac{6}{-6} > x > \frac{-6}{-6}

Simplifying the fractions, we get:

−1>x>1-1 > x > 1

Step 4: Write the Solution in Interval Notation

The solution to the inequality can be written in interval notation as (−1,1)(-1, 1). This means that the values of xx that satisfy the inequality are all the real numbers between -1 and 1, but not including -1 and 1.

Conclusion

Solving absolute value inequalities can be challenging, but with a clear understanding of the concept and a step-by-step approach, they can be tackled with ease. In this article, we solved the inequality ∣−6x+2∣<8|-6x + 2| < 8 by writing it as a double inequality, subtracting 2 from each part, dividing each part by -6, and writing the solution in interval notation. We hope that this article has provided a clear and concise guide to solving absolute value inequalities.

Common Mistakes to Avoid

When solving absolute value inequalities, there are several common mistakes to avoid. Here are a few:

  • Not writing the inequality as a double inequality: Failing to write the inequality as a double inequality can make it difficult to solve.
  • Not subtracting 2 from each part of the inequality: Failing to subtract 2 from each part of the inequality can make it difficult to isolate the term with the variable.
  • Not dividing each part of the inequality by -6: Failing to divide each part of the inequality by -6 can make it difficult to solve for xx.
  • Not writing the solution in interval notation: Failing to write the solution in interval notation can make it difficult to understand the solution.

Real-World Applications

Absolute value inequalities have many real-world applications. Here are a few:

  • Finance: In finance, absolute value inequalities can be used to model the value of a stock or a bond. For example, if the value of a stock is $100, and the absolute value of the change in value is less than 10%, then the value of the stock is still $100.
  • Science: In science, absolute value inequalities can be used to model the behavior of physical systems. For example, if the temperature of a system is 20 degrees Celsius, and the absolute value of the change in temperature is less than 5 degrees Celsius, then the temperature of the system is still 20 degrees Celsius.
  • Engineering: In engineering, absolute value inequalities can be used to model the behavior of complex systems. For example, if the speed of a car is 60 miles per hour, and the absolute value of the change in speed is less than 10 miles per hour, then the speed of the car is still 60 miles per hour.

Conclusion

Introduction

In our previous article, we solved the inequality ∣−6x+2∣<8|-6x + 2| < 8 by writing it as a double inequality, subtracting 2 from each part, dividing each part by -6, and writing the solution in interval notation. In this article, we will provide a Q&A guide to help you better understand the concept of absolute value inequalities and how to solve them.

Q: What is an absolute value inequality?

A: An absolute value inequality is an inequality that involves the absolute value of an expression. It is a type of inequality that can be challenging to solve, but with a clear understanding of the concept and a step-by-step approach, they can be tackled with ease.

Q: How do I write an absolute value inequality as a double inequality?

A: To write an absolute value inequality as a double inequality, you need to isolate the absolute value expression and then write two separate inequalities. For example, if you have the inequality ∣x−3∣<5|x - 3| < 5, you can write it as −5<x−3<5-5 < x - 3 < 5.

Q: What is the difference between a double inequality and a single inequality?

A: A double inequality is an inequality that has two parts: one part that is less than or equal to a certain value, and another part that is greater than or equal to the same value. A single inequality, on the other hand, is an inequality that has only one part. For example, the inequality −5<x<5-5 < x < 5 is a single inequality, while the inequality −5<x−3<5-5 < x - 3 < 5 is a double inequality.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to follow these steps:

  1. Write the inequality as a double inequality.
  2. Subtract the constant term from each part of the inequality.
  3. Divide each part of the inequality by the coefficient of the variable.
  4. Write the solution in interval notation.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality in a compact and concise form. It is a way of representing a range of values using a specific notation. For example, the solution to the inequality −5<x<5-5 < x < 5 can be written in interval notation as (−5,5)(-5, 5).

Q: How do I write the solution to an absolute value inequality in interval notation?

A: To write the solution to an absolute value inequality in interval notation, you need to follow these steps:

  1. Identify the two critical points of the inequality.
  2. Write the solution as a range of values between the two critical points.
  3. Use the correct notation to represent the solution.

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:

  • Not writing the inequality as a double inequality.
  • Not subtracting the constant term from each part of the inequality.
  • Not dividing each part of the inequality by the coefficient of the variable.
  • Not writing the solution in interval notation.

Q: How do I apply absolute value inequalities in real-world problems?

A: Absolute value inequalities can be applied in a variety of real-world problems, including finance, science, and engineering. For example, in finance, absolute value inequalities can be used to model the value of a stock or a bond. In science, absolute value inequalities can be used to model the behavior of physical systems. In engineering, absolute value inequalities can be used to model the behavior of complex systems.

Conclusion

In conclusion, solving absolute value inequalities can be challenging, but with a clear understanding of the concept and a step-by-step approach, they can be tackled with ease. In this article, we provided a Q&A guide to help you better understand the concept of absolute value inequalities and how to solve them. We hope that this article has provided a clear and concise guide to solving absolute value inequalities.