📚 Understanding Absolute Value Inequalities
Absolute value inequalities involve expressions within absolute value bars and an inequality symbol. The expressions $|ax + b| < c$ and $|ax + b| \le c$ represent slightly different solution sets due to the strictness of the inequality.
🕰️ Historical Context
The concept of absolute value has been used for centuries, but its formalization and use in inequalities became prominent in the development of modern algebra in the 19th century. Mathematicians sought to define distances and magnitudes rigorously, leading to the widespread adoption of absolute value notation.
📌 Key Principles
- 🔍 Definition of Absolute Value: The absolute value of a number $x$, denoted as $|x|$, is its distance from zero. It is always non-negative. Mathematically, $|x| = x$ if $x \ge 0$, and $|x| = -x$ if $x < 0$.
- 💡 Solving $|ax + b| < c$: This inequality means that the expression $ax + b$ must be within $c$ units of zero. It translates to two inequalities: $-c < ax + b < c$. The solution set includes all values of $x$ that satisfy both inequalities simultaneously, excluding the endpoints.
- 📝 Solving $|ax + b| \le c$: This inequality means that the expression $ax + b$ must be within $c$ units of zero, including the endpoints. It translates to two inequalities: $-c \le ax + b \le c$. The solution set includes all values of $x$ that satisfy both inequalities, including the endpoints.
- 🧮 Graphical Representation: On a number line, the solution to $|ax + b| < c$ is an open interval, while the solution to $|ax + b| \le c$ is a closed interval.
- 🧪 Impact of 'Strictness': The key difference lies in whether the endpoints are included in the solution set. '$<$' excludes the endpoints, while '$\le$' includes them.
🌍 Real-world Examples
Consider a scenario where a machine needs to operate within a certain temperature range to function correctly.
- 🌡️ Example 1: A machine operates correctly if its internal temperature $T$ satisfies $|T - 25| < 5$. This means the temperature must be strictly between 20 and 30 degrees Celsius (i.e., $20 < T < 30$). If the temperature reaches exactly 20 or 30 degrees, the machine might malfunction.
- ⚙️ Example 2: A manufacturing process requires a part to be within a certain tolerance. The length $L$ of a part must satisfy $|L - 10| \le 0.1$. This means the length can be between 9.9 and 10.1 units, inclusive (i.e., $9.9 \le L \le 10.1$). Parts with lengths of exactly 9.9 or 10.1 are still acceptable.
✔️ Conclusion
The distinction between $|ax + b| < c$ and $|ax + b| \le c$ hinges on whether the endpoints are included in the solution. '$<$' provides a strict inequality, excluding endpoints, while '$\le$' includes them. Understanding this difference is crucial for accurately solving inequalities and interpreting their solutions in real-world contexts.