Understanding open intervals in absolute value inequalities |ax + b| > c

📚 Understanding Open Intervals in Absolute Value Inequalities

Let's delve into absolute value inequalities of the form $|ax + b| > c$, where $a$, $b$, and $c$ are real numbers, and $c > 0$. This type of inequality represents all values of $x$ for which the distance between $ax + b$ and 0 is greater than $c$. The solution set will typically be an open interval or the union of two open intervals.

📜 A Brief History and Background

The concept of absolute value has been around for centuries, finding roots in number theory and analysis. The formalization of inequalities, including those involving absolute values, came with the development of modern algebra and calculus. Solving these inequalities is crucial in various fields, including engineering, physics, and computer science.

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

• ➕ Splitting the Inequality: The inequality $|ax + b| > c$ is equivalent to two separate inequalities: $ax + b > c$ or $ax + b < -c$. This is because the expression inside the absolute value can be either positive or negative, but its distance from zero must be greater than $c$.
• ➗ Solving for $x$: Solve each inequality separately. For $ax + b > c$, subtract $b$ from both sides to get $ax > c - b$, then divide by $a$. Remember that if $a$ is negative, you must flip the inequality sign. Similarly, for $ax + b < -c$, subtract $b$ to get $ax < -c - b$, and again divide by $a$, flipping the sign if $a$ is negative.
• 📈 Open Interval Notation: The solution to each inequality will be an open interval. For example, if $x > d$, the interval is $(d, \infty)$, and if $x < e$, the interval is $(-\infty, e)$. The final solution is the union of these intervals.
• 🤝 Union of Intervals: Since we have 'or' connecting the two inequalities, the overall solution is the union of the two open intervals. This means any $x$ that satisfies either inequality is part of the solution.

💡 Real-World Examples

Let's work through some examples to solidify the concept:

1. Example 1: Solve $|2x - 1| > 3$.
   We have two inequalities: $2x - 1 > 3$ or $2x - 1 < -3$.
   Solving $2x - 1 > 3$: $2x > 4$, so $x > 2$.
   Solving $2x - 1 < -3$: $2x < -2$, so $x < -1$.
   The solution is $x < -1$ or $x > 2$, which in interval notation is $(-\infty, -1) \cup (2, \infty)$.
2. Example 2: Solve $|-3x + 6| > 9$.
   We have two inequalities: $-3x + 6 > 9$ or $-3x + 6 < -9$.
   Solving $-3x + 6 > 9$: $-3x > 3$, so $x < -1$ (note the sign flip).
   Solving $-3x + 6 < -9$: $-3x < -15$, so $x > 5$ (note the sign flip).
   The solution is $x < -1$ or $x > 5$, which in interval notation is $(-\infty, -1) \cup (5, \infty)$.
3. Example 3: Solve $|x + 2| > 0$.
   We have two inequalities: $x + 2 > 0$ or $x + 2 < 0$.
   Solving $x + 2 > 0$: $x > -2$.
   Solving $x + 2 < 0$: $x < -2$.
   The solution is all real numbers except $x = -2$, which in interval notation is $(-\infty, -2) \cup (-2, \infty)$.

📝 Conclusion

Understanding open intervals in absolute value inequalities involves splitting the absolute value into two separate inequalities and solving each independently. The final solution is the union of the resulting open intervals. Remember to flip the inequality sign when dividing by a negative number. With practice, these problems become much easier to solve!