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

Let's dive into the world of absolute value inequalities! This guide will walk you through solving the inequality $-5|x-2|+2 eq -13$ and expressing the solution in interval notation. Don't worry, we'll break it down into easy-to-follow steps. Grab your favorite beverage, and let's get started!

Understanding Absolute Value Inequalities

Before we jump into the specific problem, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero. For example, $|3| = 3$ and $|-3| = 3$. When we're dealing with inequalities involving absolute values, we're essentially looking for a range of values that satisfy a certain condition related to distance.

Absolute value inequalities often require us to consider two separate cases because the expression inside the absolute value can be either positive or negative. This is crucial for correctly solving the inequality and expressing the solution in interval notation.

Now, inequalities are mathematical statements that compare two expressions using symbols like $<$, $>$, $\leq$, or $\geq$. When you combine absolute values with inequalities, you need to be extra careful to handle the two potential scenarios arising from the absolute value. These scenarios are when the expression inside the absolute value is positive or negative, which leads to different equations to solve.

To successfully solve absolute value inequalities, remember to isolate the absolute value expression first. Then, consider both positive and negative cases, solve each separately, and combine the solutions appropriately. Finally, represent your solution in interval notation to clearly define the range of values that satisfy the original inequality. Understanding these fundamental concepts will set you up for success in tackling more complex problems!

Step-by-Step Solution

1. Isolate the Absolute Value

Our first goal is to isolate the absolute value term. We start with the inequality:

$$-5|x-2|+2 eq -13$$

Subtract 2 from both sides:

$$-5|x-2| eq -15$$

Now, divide both sides by -5. Remember that when we divide or multiply an inequality by a negative number, we need to flip the inequality sign:

$$|x-2| < 3$$

2. Break into Two Cases

Since we're dealing with an absolute value, we need to consider two cases:

Case 1: The expression inside the absolute value is positive or zero:

$$x-2 < 3$$

Case 2: The expression inside the absolute value is negative:

-(x-2) < 3$ which simplifies to $x-2 > -3

3. Solve Each Case

Let's solve each case separately.

Case 1:

$$x-2 < 3$$

Add 2 to both sides:

$$x < 5$$

Case 2:

$$x-2 > -3$$

Add 2 to both sides:

$$x > -1$$

4. Combine the Solutions

We have two inequalities: $x < 5$ and $x > -1$. We need to find the values of $x$ that satisfy both of these inequalities. In other words, $x$ must be greater than -1 and less than 5.

5. Express in Interval Notation

Now, let's express this solution in interval notation. Since $x$ is strictly greater than -1 and strictly less than 5, we use parentheses to denote that -1 and 5 are not included in the solution set.

The interval notation for $x > -1$ and $x < 5$ is:

$$(-1, 5)$$

Common Mistakes to Avoid

• Forgetting to Flip the Inequality Sign: When dividing or multiplying by a negative number, always remember to flip the inequality sign. Failing to do so will lead to an incorrect solution.
• Incorrectly Handling Absolute Value: Remember to consider both positive and negative cases for the expression inside the absolute value. Many mistakes occur when only one case is considered.
• Not Isolating the Absolute Value First: Always isolate the absolute value expression before breaking the problem into cases. This ensures you're working with the correct inequality.
• Misinterpreting Interval Notation: Be careful with using parentheses and brackets. Parentheses indicate that the endpoint is not included, while brackets indicate that it is. Double-check your solution to ensure you're using the correct notation.

Practice Problems

Want to test your understanding? Try these practice problems:

1. Solve $|2x + 1| < 5$ and express the solution in interval notation.
2. Solve $-3|x - 4| + 7 > 1$ and express the solution in interval notation.
3. Solve $|5x - 3| eq 2$ and express the solution in interval notation.

Conclusion

Alright, guys, that's how you solve the inequality $-5|x-2|+2 eq -13$ and express the solution in interval notation. Remember to isolate the absolute value, consider both cases, and combine your solutions carefully. With a bit of practice, you'll be a pro at solving absolute value inequalities! Keep up the great work, and don't hesitate to ask if you have any questions. Happy solving!

By following these steps and avoiding common mistakes, you'll be well-equipped to tackle any absolute value inequality that comes your way. Remember, practice makes perfect, so keep solving problems and refining your skills! If you found this guide helpful, share it with your friends and classmates. Let's conquer math together!

Key Takeaways:

• Isolate: Always isolate the absolute value term first.
• Cases: Consider both positive and negative cases.
• Flip: Remember to flip the inequality sign when multiplying or dividing by a negative number.
• Notation: Use the correct interval notation to express your solution.

With these tips in mind, you'll be solving absolute value inequalities like a champ in no time! Keep practicing, and you'll master this topic in no time. If you get stuck, remember to review the steps and examples provided in this guide. Good luck, and happy problem-solving!