Graphing the solution set for |ax + b| < c absolute value inequalities

📚 Understanding Absolute Value Inequalities

Absolute value inequalities involve expressions of the form $|ax + b| < c$ (or variations with $\leq$, $>$, or $\geq$). The absolute value of a number represents its distance from zero. Solving these inequalities means finding all values of $x$ that satisfy the given condition.

📜 Historical Context

The concept of absolute value has been used implicitly for centuries, but its formal notation and systematic study emerged in the 19th century with the development of real analysis. Mathematicians like Karl Weierstrass contributed to the rigorous definition and application of absolute value in various mathematical contexts.

🔑 Key Principles for Solving $|ax + b| < c$

• 📏 Definition of Absolute Value: The absolute value $|x|$ is defined as $x$ if $x \geq 0$ and $-x$ if $x < 0$.
• ➗ Splitting the Inequality: $|ax + b| < c$ is equivalent to two inequalities: $-c < ax + b < c$.
• ➕ Isolating $x$: Solve the compound inequality by isolating $x$ in the middle.
• 📈 Graphing the Solution Set: Represent the solution on a number line. For '$<$' or '$>$', use open circles; for '$\leq$' or '$\geq$', use closed circles.

✏️ Step-by-Step Solution

Let's solve $|2x - 1| < 5$:

1. Split the inequality: $-5 < 2x - 1 < 5$
2. Add 1 to all parts: $-5 + 1 < 2x - 1 + 1 < 5 + 1$, which simplifies to $-4 < 2x < 6$
3. Divide all parts by 2: $\frac{-4}{2} < \frac{2x}{2} < \frac{6}{2}$, which simplifies to $-2 < x < 3$
4. Graph the solution: On a number line, place open circles at -2 and 3, and shade the region between them.

💡 Real-World Examples

• 🌡️ Temperature Control: A thermostat is set to maintain a temperature within a certain range. For example, $|T - 70| < 3$ means the temperature $T$ should be between 67 and 73 degrees.
• ⚙️ Manufacturing Tolerances: In manufacturing, parts must be made within a certain tolerance. If a part should be 10 cm long with a tolerance of 0.1 cm, then $|L - 10| < 0.1$, where $L$ is the actual length of the part.

📊 Practice Quiz

Solve and graph the following inequalities:

1. $|x - 3| < 2$
2. $|2x + 1| \leq 5$
3. $|3x - 4| > 8$

📝 Solutions to Practice Quiz

1. $|x - 3| < 2$ => $1 < x < 5$
2. $|2x + 1| \leq 5$ => $-3 \leq x \leq 2$
3. $|3x - 4| > 8$ => $x < -\frac{4}{3}$ or $x > 4$

🔑 Conclusion

Graphing the solution set for absolute value inequalities involves understanding the definition of absolute value, splitting the inequality into two separate inequalities, solving for $x$, and representing the solution on a number line. By following these steps, you can confidently solve and graph these inequalities. Keep practicing, and you'll master this concept in no time!