Step-by-step guide to solving |ax + b| < c inequalities in Algebra 1

📚 Understanding Absolute Value Inequalities

Absolute value inequalities might seem intimidating, but they break down into manageable parts. The key is understanding what absolute value means: the distance from zero. When you see $|ax + b| < c$, it means the expression $ax + b$ must be within $c$ units of zero. This translates into two separate inequalities that you need to solve.

📜 A Little Background

The concept of absolute value has been around for a long time, though its modern notation was formalized in the 19th century. It’s crucial in various branches of mathematics and physics for defining distance and magnitude without considering direction or sign.

🔑 Key Principles

• 📏 Definition of Absolute Value: The absolute value of a number $x$, denoted as $|x|$, is its distance from 0. Formally, $|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$
• ➕ Splitting the Inequality: When solving $|ax + b| < c$, you need to split it into two separate inequalities:
  • $ax + b < c$
  • $ax + b > -c$
• ✍️ Solving the Inequalities: Solve each inequality independently using standard algebraic techniques. Remember to perform the same operation on both sides to maintain the inequality.
• 🤝 Combining the Solutions: The solution to the absolute value inequality is the intersection (AND) of the solutions to the two separate inequalities.
• 💡 Graphical Interpretation: You can visualize the solution on a number line. The solution represents the values of $x$ that satisfy both inequalities simultaneously.

🪜 Step-by-Step Solution

Let's break down how to solve $|ax + b| < c$:

1. 📝 Isolate the Absolute Value: Ensure the absolute value expression is isolated on one side of the inequality. In this case, it already is.
2. ➗ Split into Two Inequalities: Create two inequalities:
• $ax + b < c$
• $ax + b > -c$
3. ➕ Solve Each Inequality:
   1. Subtract $b$ from all parts of each inequality:
      • $ax < c - b$
      • $ax > -c - b$
   2. Divide all parts by $a$. Important: If $a$ is negative, remember to flip the inequality signs!
      • If $a > 0$: $x < \frac{c - b}{a}$ and $x > \frac{-c - b}{a}$
      • If $a < 0$: $x > \frac{c - b}{a}$ and $x < \frac{-c - b}{a}$
4. 📈 Write the Solution Set: The solution set will be the intersection of the two solutions. This can be written as an inequality or in interval notation.

🧮 Real-World Examples

Let's walk through a couple of examples:

1. Example 1: Solve $|2x + 1| < 5$
   1. Split: $2x + 1 < 5$ and $2x + 1 > -5$
   2. Solve:
      • $2x < 4$ => $x < 2$
      • $2x > -6$ => $x > -3$
   3. Solution: $-3 < x < 2$ or $x \in (-3, 2)$
2. Example 2: Solve $|-3x + 6| < 9$
   1. Split: $-3x + 6 < 9$ and $-3x + 6 > -9$
   2. Solve:
      • $-3x < 3$ => $x > -1$ (Notice the flip because we divide by -3)
      • $-3x > -15$ => $x < 5$ (Notice the flip because we divide by -3)
   3. Solution: $-1 < x < 5$ or $x \in (-1, 5)$

📝 Practice Quiz

Test your understanding!

1. Solve: $|x - 3| < 2$
2. Solve: $|2x + 4| < 6$
3. Solve: $|-x + 1| < 3$
4. Solve: $|3x - 5| < 4$
5. Solve: $|-2x - 2| < 8$

Answers:

1. $1 < x < 5$
2. $-5 < x < 1$
3. $-2 < x < 4$
4. $\frac{1}{3} < x < 3$
5. $-5 < x < 3$

📊 Conclusion

Solving absolute value inequalities involves understanding the definition of absolute value and then breaking the problem into two separate inequalities. With practice, these types of problems become much easier. Keep practicing and you'll master them in no time! 💪