Solve The Inequality: ${ |3x - 9| \ \textless \ 1 }$

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Introduction

In this article, we will delve into the world of inequalities and focus on solving the inequality |3x - 9| < 1. Inequalities are mathematical expressions that compare two values, and they can be either greater than, less than, greater than or equal to, or less than or equal to. In this case, we are dealing with an absolute value inequality, which means that the expression inside the absolute value bars can be either positive or negative.

Understanding Absolute Value Inequalities

Absolute value inequalities involve the absolute value of an expression, which is the distance of the expression from zero on the number line. The absolute value of a number is always non-negative, and it can be thought of as the distance of the number from zero. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

When dealing with absolute value inequalities, we need to consider two cases: one where the expression inside the absolute value bars is positive, and one where it is negative. This is because the absolute value of a negative number is always positive, and vice versa.

Solving the Inequality |3x - 9| < 1

To solve the inequality |3x - 9| < 1, we need to consider two cases:

Case 1: 3x - 9 ≥ 0

In this case, the expression inside the absolute value bars is non-negative, and we can simply remove the absolute value bars. The inequality becomes:

3x - 9 < 1

To solve for x, we can add 9 to both sides of the inequality:

3x < 10

Next, we can divide both sides of the inequality by 3:

x < 10/3

x < 3.33

Case 2: 3x - 9 < 0

In this case, the expression inside the absolute value bars is negative, and we need to multiply it by -1 to make it positive. The inequality becomes:

-(3x - 9) < 1

To simplify the inequality, we can distribute the negative sign to the terms inside the parentheses:

-3x + 9 < 1

Next, we can subtract 9 from both sides of the inequality:

-3x < -8

To solve for x, we can divide both sides of the inequality by -3. However, because we are dividing by a negative number, we need to flip the direction of the inequality:

x > 8/3

x > 2.67

Combining the Two Cases

Now that we have solved the inequality for both cases, we can combine the two solutions to get the final answer. The solution to the inequality |3x - 9| < 1 is:

-2.67 < x < 3.33

This means that the value of x must be between -2.67 and 3.33, exclusive.

Conclusion

In this article, we have solved the inequality |3x - 9| < 1 by considering two cases: one where the expression inside the absolute value bars is non-negative, and one where it is negative. We have used algebraic manipulations to simplify the inequality and solve for x. The final solution is -2.67 < x < 3.33, which means that the value of x must be between -2.67 and 3.33, exclusive.

Frequently Asked Questions

  • What is an absolute value inequality? An absolute value inequality is a mathematical expression that compares two values, and it can be either greater than, less than, greater than or equal to, or less than or equal to. In this case, we are dealing with an absolute value inequality of the form |3x - 9| < 1.
  • How do I solve an absolute value inequality? To solve an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value bars is non-negative, and one where it is negative. You can then use algebraic manipulations to simplify the inequality and solve for x.
  • What is the final solution to the inequality |3x - 9| < 1? The final solution to the inequality |3x - 9| < 1 is -2.67 < x < 3.33, which means that the value of x must be between -2.67 and 3.33, exclusive.

References

Further Reading

Introduction

In our previous article, we solved the inequality |3x - 9| < 1 by considering two cases: one where the expression inside the absolute value bars is non-negative, and one where it is negative. We used algebraic manipulations to simplify the inequality and solve for x. In this article, we will answer some frequently asked questions about solving absolute value inequalities.

Q&A

Q: What is an absolute value inequality?

A: An absolute value inequality is a mathematical expression that compares two values, and it can be either greater than, less than, greater than or equal to, or less than or equal to. In this case, we are dealing with an absolute value inequality of the form |3x - 9| < 1.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value bars is non-negative, and one where it is negative. You can then use algebraic manipulations to simplify the inequality and solve for x.

Q: What is the final solution to the inequality |3x - 9| < 1?

A: The final solution to the inequality |3x - 9| < 1 is -2.67 < x < 3.33, which means that the value of x must be between -2.67 and 3.33, exclusive.

Q: Can I use a calculator to solve absolute value inequalities?

A: Yes, you can use a calculator to solve absolute value inequalities. However, it's always a good idea to check your work by hand to make sure you understand the steps involved.

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 bars is non-negative, and one where it is negative. To determine which case to use, you can set up a sign chart or use a number line to visualize the expression.

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. However, it's often easier to use other methods, such as substitution or elimination, to solve systems of equations.

Q: How do I graph absolute value inequalities on a number line?

A: To graph an absolute value inequality on a number line, you need to identify the critical points of the inequality and then plot the corresponding points on the number line. You can then use a test point to determine which intervals satisfy the inequality.

Conclusion

In this article, we have answered some frequently asked questions about solving absolute value inequalities. We have discussed the basics of absolute value inequalities, how to solve them, and how to graph them on a number line. We hope this article has been helpful in clarifying any confusion you may have had about solving absolute value inequalities.

Frequently Asked Questions

  • What is an absolute value inequality?
  • How do I solve an absolute value inequality?
  • What is the final solution to the inequality |3x - 9| < 1?
  • Can I use a calculator to solve absolute value inequalities?
  • How do I know which case to use when solving an absolute value inequality?
  • Can I use absolute value inequalities to solve systems of equations?
  • How do I graph absolute value inequalities on a number line?

References

Further Reading