Solving Absolute Value Inequalities: A Step-by-Step Guide

Understanding and solving absolute value inequalities can seem daunting at first, but with a clear approach, it becomes quite manageable. This guide will walk you through the process, focusing on how to correctly identify the compound inequality that corresponds to a given absolute value inequality. We'll use the example of $|3x - 5| > 10$ to illustrate the key concepts and steps involved.

Understanding Absolute Value Inequalities

At its core, the absolute value of a number represents its distance from zero on the number line. Therefore, an absolute value inequality like $|3x - 5| > 10$ is asking: for what values of x is the distance between 3x - 5 and zero greater than 10? This is a crucial concept to grasp because it directly leads to the understanding of compound inequalities.

Why Compound Inequalities? Absolute value inequalities translate into compound inequalities because there are two scenarios to consider. If the expression inside the absolute value bars is greater than a certain positive number, it means the expression is far to the right of zero. Conversely, if the expression is "less than" the negative of that number, it means the expression is far to the left of zero. This "either/or" situation is what necessitates the use of compound inequalities involving "or."

Key Principles to Remember When dealing with absolute value inequalities, keep these principles in mind:

• $|x| < a$ translates to $-a < x < a$ (a bounded interval).
• $|x| > a$ translates to $x < -a$ or $x > a$ (two unbounded intervals).

These principles are the foundation for converting absolute value inequalities into a form we can solve.

Deconstructing the Inequality $|3x - 5| > 10$

Now, let's apply these principles to our specific example: $|3x - 5| > 10$. This inequality tells us that the distance between 3x - 5 and zero is greater than 10. This leads to two separate cases:

1. Case 1: 3x - 5 is greater than 10. This means 3x - 5 > 10. In this case, the expression 3x - 5 is far to the right of zero.
2. Case 2: 3x - 5 is less than -10. This means 3x - 5 < -10. Here, the expression 3x - 5 is far to the left of zero.

These two cases form the compound inequality that represents the original absolute value inequality. The word "or" connects these cases because either condition satisfies the original inequality. It's crucial to recognize that both conditions cannot be true simultaneously; 3x - 5 can't be both greater than 10 and less than -10 at the same time.

Identifying the Correct Compound Inequality

Based on our analysis, the compound inequality equivalent to $|3x - 5| > 10$ is:

3x - 5 < -10 or 3x - 5 > 10

Let's examine why the other options are incorrect:

• A. -10 < |3x - 5| < 10: This inequality implies that the absolute value of 3x - 5 is less than 10, which is the opposite of what our original inequality states.
• B. 3x - 5 > -10 or 3x - 5 < 10: While these are inequalities, they represent the solution to $|3x - 5| < 10$, not $|3x - 5| > 10$. This is a common mistake, so pay close attention to the direction of the original inequality.
• D. -10 < 3x - 5 < 10: This inequality represents a bounded interval, implying that 3x - 5 is between -10 and 10. Again, this corresponds to $|3x - 5| < 10$, not our original inequality.

Therefore, the correct answer is C: 3x - 5 < -10 or 3x - 5 > 10.

Solving the Compound Inequality

To fully understand the solution, let's solve the compound inequality we've identified. We solve each inequality separately:

1. Solve 3x - 5 < -10

• Add 5 to both sides: 3x < -5
• Divide both sides by 3: x < -5/3

2. Solve 3x - 5 > 10

• Add 5 to both sides: 3x > 15
• Divide both sides by 3: x > 5

Thus, the solution to the inequality $|3x - 5| > 10$ is x < -5/3 or x > 5. This means any value of x less than -5/3 or greater than 5 will satisfy the original inequality. This solution set consists of two unbounded intervals on the number line.

Visualizing the Solution

A helpful way to solidify your understanding is to visualize the solution on a number line. Mark -5/3 and 5 on the number line. Since our solution includes x < -5/3 and x > 5, we'll shade the regions to the left of -5/3 and to the right of 5. We use open circles at -5/3 and 5 to indicate that these points are not included in the solution (because the original inequality is "greater than," not "greater than or equal to"). Visualizing the solution set can help prevent errors and reinforce the concept of unbounded intervals.

Common Pitfalls to Avoid

When working with absolute value inequalities, there are a few common mistakes to watch out for:

• Incorrectly interpreting the direction of the inequality: Forgetting that |x| > a translates to an "or" compound inequality and confusing it with the "and" inequality that arises from |x| < a.
• Not considering both cases: Failing to split the absolute value inequality into its two constituent inequalities.
• Making algebraic errors: Careless mistakes in solving the individual inequalities can lead to an incorrect solution set. Double-check your steps!
• Forgetting to reverse the inequality sign: When multiplying or dividing by a negative number, remember to flip the direction of the inequality sign.

Practice Makes Perfect

The key to mastering absolute value inequalities is practice. Work through various examples, paying close attention to the steps involved in converting the absolute value inequality into a compound inequality and solving it. The more you practice, the more comfortable and confident you'll become.

Conclusion

Solving absolute value inequalities involves understanding the concept of distance from zero and correctly applying the rules for converting them into compound inequalities. By carefully considering the two cases that arise from the absolute value, you can accurately identify the equivalent compound inequality and solve for the variable. Remember to visualize the solution set and practice regularly to avoid common pitfalls. With a solid understanding of these principles, you'll be well-equipped to tackle any absolute value inequality that comes your way.

For further learning and practice on inequalities, you can visit Khan Academy's Algebra I section on inequalities.