Integrated Math 2: Absolute Value Inequalities: Explained with Examples

Understanding Absolute Value Inequalities 🤔

Absolute value inequalities involve expressions like |x| < a or |x| > a, where 'x' is a variable and 'a' is a constant. Solving these inequalities requires considering two cases due to the nature of absolute value, which represents the distance from zero.

Basic Principles 🔑

• |x| < a: This means x is within a distance of 'a' from zero. Therefore, -a < x < a.
• |x| > a: This means x is farther than a distance of 'a' from zero. Therefore, x < -a or x > a.

Step-by-Step Guide with Examples 🚀

Example 1: Solving |x - 2| < 3

1. Set up two inequalities:
   • x - 2 < 3
   • -(x - 2) < 3 which simplifies to x - 2 > -3
2. Solve the first inequality:

   x - 2 < 3

   Add 2 to both sides:

   x < 5

3. Solve the second inequality:

   x - 2 > -3

   Add 2 to both sides:

   x > -1

4. Combine the solutions:

   -1 < x < 5
5. Solution Set:

   The solution set is all x such that -1 < x < 5. In interval notation: (-1, 5)

Example 2: Solving |2x + 1| > 5

1. Set up two inequalities:
   • 2x + 1 > 5
   • -(2x + 1) > 5 which simplifies to 2x + 1 < -5
2. Solve the first inequality:

   2x + 1 > 5

   Subtract 1 from both sides:

   2x > 4

   Divide by 2:

   x > 2

3. Solve the second inequality:

   2x + 1 < -5

   Subtract 1 from both sides:

   2x < -6

   Divide by 2:

   x < -3

4. Combine the solutions:

   x < -3 or x > 2

5. Solution Set:

   The solution set is all x such that x < -3 or x > 2. In interval notation: (-∞, -3) ∪ (2, ∞)

Advanced Example with Fractions and Negatives 🧠

Example 3: Solving |(3x - 2)/4| ≤ 2

1. Set up two inequalities:
   • (3x - 2)/4 ≤ 2
   • -(3x - 2)/4 ≤ 2 which simplifies to (3x - 2)/4 ≥ -2
2. Solve the first inequality:

   (3x - 2)/4 ≤ 2

   Multiply both sides by 4:

   3x - 2 ≤ 8

   Add 2 to both sides:

   3x ≤ 10

   Divide by 3:

   x ≤ 10/3

3. Solve the second inequality:

   (3x - 2)/4 ≥ -2

   Multiply both sides by 4:

   3x - 2 ≥ -8

   Add 2 to both sides:

   3x ≥ -6

   Divide by 3:

   x ≥ -2

4. Combine the solutions:

   -2 ≤ x ≤ 10/3

5. Solution Set:

   The solution set is all x such that -2 ≤ x ≤ 10/3. In interval notation: [-2, 10/3]

Tips for Success ✅

• Isolate the Absolute Value: Always isolate the absolute value expression before splitting into two cases.
• Remember Two Cases: One case for the positive value and one for the negative value.
• Pay Attention to the Inequality Sign: The direction of the inequality sign matters!
• Check Your Solutions: Substitute your solutions back into the original inequality to verify.

Practice Problems ✍️

Try solving these on your own:

• |x + 3| < 5
• |3x - 1| ≥ 2
• |(2x + 4)/2| > 1

By following these steps and practicing regularly, you'll master absolute value inequalities in Integrated Math 2!