Solving Inequalities With Absolute Value Worksheet

Solving Inequalities with Absolute Value: A Comprehensive Guide

Solving inequalities involving absolute value can seem daunting at first, but with a systematic approach and a solid understanding of the underlying principles, you can master this crucial skill. This comprehensive guide will walk you through various techniques, providing clear explanations and examples to help you confidently tackle any absolute value inequality problem. We'll delve into the core concepts, explore different solution methods, and offer practical tips to ensure you achieve mastery. This guide serves as a virtual worksheet, allowing you to practice along the way.

Understanding Absolute Value

Before diving into inequalities, let's solidify our understanding of absolute value. The absolute value of a number is its distance from zero on the number line. Therefore, it's always non-negative. We represent the absolute value of a number x as |x|.

• |x| = x if x ≥ 0 (The absolute value of a non-negative number is the number itself.)
• |x| = -x if x < 0 (The absolute value of a negative number is its opposite.)

For example:

• |5| = 5
• |-5| = 5
• |0| = 0

Types of Absolute Value Inequalities

Absolute value inequalities fall into two main categories:

• |x| < a (where 'a' is a positive number)
• |x| > a (where 'a' is a positive number)

Let's examine each type in detail.

Solving Inequalities of the Form |x| < a

Inequalities of the form |x| < a represent all values of x whose distance from zero is less than a. This translates to a compound inequality:

-a < x < a

This means x is greater than -a and less than a. The solution is an interval.

Example 1: Solve |x| < 3

Solution: This inequality is equivalent to -3 < x < 3. The solution is the interval (-3, 3). This means x can be any number between -3 and 3, excluding -3 and 3.

Example 2: Solve |x + 2| < 5

Solution: We treat the expression inside the absolute value as our 'x'. Therefore, we have:

-5 < x + 2 < 5

Subtract 2 from all parts of the inequality:

-5 - 2 < x < 5 - 2

-7 < x < 3

The solution is the interval (-7, 3).

Solving Inequalities of the Form |x| > a

Inequalities of the form |x| > a represent all values of x whose distance from zero is greater than a. This translates to two separate inequalities:

x > a or x < -a

The solution consists of two intervals.

Example 3: Solve |x| > 2

Solution: This inequality is equivalent to x > 2 or x < -2. The solution consists of two intervals: (-∞, -2) and (2, ∞).

Example 4: Solve |2x - 1| > 5

Solution: We have two inequalities:

2x - 1 > 5 or 2x - 1 < -5

Solve each inequality separately:

• 2x - 1 > 5: Add 1 to both sides: 2x > 6. Divide by 2: x > 3
• 2x - 1 < -5: Add 1 to both sides: 2x < -4. Divide by 2: x < -2

The solution is x > 3 or x < -2. This can be represented as the intervals (-∞, -2) and (3, ∞).

Solving More Complex Absolute Value Inequalities

More complex inequalities might involve multiple absolute values or require additional algebraic manipulation before applying the principles outlined above. Let's consider some examples.

Example 5: Solve |x - 1| + 2 < 5

Solution: First, isolate the absolute value term:

|x - 1| < 5 - 2

|x - 1| < 3

Now, we can apply the rule for |x| < a:

-3 < x - 1 < 3

Add 1 to all parts of the inequality:

-3 + 1 < x < 3 + 1

-2 < x < 4

The solution is the interval (-2, 4).

Example 6: Solve |x + 2| ≥ |x - 3|

Solution: This type of inequality requires a slightly different approach. We need to consider different cases based on the signs of (x + 2) and (x - 3). However, a simpler approach is to square both sides (since both sides are non-negative):

(x + 2)² ≥ (x - 3)²

x² + 4x + 4 ≥ x² - 6x + 9

Simplify by subtracting x² from both sides:

4x + 4 ≥ -6x + 9

10x ≥ 5

x ≥ 1/2

Therefore, the solution is the interval [1/2, ∞).

Graphical Representation of Solutions

Visualizing the solution sets graphically can enhance understanding. For inequalities of the form |x| < a, the solution is an interval on the number line. For inequalities of the form |x| > a, the solution consists of two separate intervals. You can represent these intervals using open or closed circles (depending on whether the endpoints are included) and shading the regions representing the solution.

Common Mistakes to Avoid

Several common pitfalls can lead to incorrect solutions. Here are some points to keep in mind:

• Incorrectly handling the 'or' condition: When solving inequalities of the form |x| > a, remember it's an 'or' condition, not an 'and' condition. Both inequalities must be considered separately.
• Forgetting to check endpoints: Always check the endpoints of your solution intervals to ensure they satisfy the original inequality.
• Incorrect algebraic manipulations: Pay close attention to the order of operations and ensure you're manipulating the inequality correctly (adding, subtracting, multiplying, or dividing).
• Not isolating the absolute value: Before applying the rules for solving absolute value inequalities, make sure the absolute value term is isolated on one side of the inequality.

Practice Problems: Your Personal Worksheet

To solidify your understanding, try solving these problems:

1. |x - 5| < 2
2. |2x + 1| ≥ 7
3. |3x - 4| < 8
4. |x + 3| + 1 > 6
5. |2x - 5| ≤ |x + 1|
6. |x - 2| > |x + 4|
7. |x² - 4| < 5
8. |1/(x-2)| > 3

Remember to follow the steps outlined in this guide. Check your answers carefully and revisit the examples if you encounter any difficulties. Practice is key to mastering absolute value inequalities. Consistent effort will ensure a strong grasp of this crucial mathematical concept. By applying the techniques and avoiding the common mistakes discussed, you'll confidently solve a wide range of absolute value inequality problems. Remember to visualize your solutions graphically for a more complete understanding. Good luck!