What are absolute value inequalities of the form |ax + b| > c?

📚 Understanding Absolute Value Inequalities of the Form $|ax + b| > c$

Absolute value inequalities of the form $|ax + b| > c$ represent all values of $x$ for which the distance between $ax + b$ and $0$ is greater than $c$. Solving these inequalities involves considering two separate cases, which arise from the definition of absolute value.

📜 History and Background

The concept of absolute value has been used implicitly for centuries, but its formalization came with the development of modern algebra and analysis. Inequalities, in general, have been studied since ancient times, with significant contributions from Greek mathematicians. The combination of absolute value and inequalities became crucial in advanced mathematics, especially in calculus and real analysis, to define concepts like limits and continuity rigorously.

🔑 Key Principles

• 📏 Definition of Absolute Value: The absolute value of a number $x$, denoted as $|x|$, is its distance from $0$ on the number line. Mathematically, $|x| = x$ if $x \geq 0$, and $|x| = -x$ if $x < 0$.
• ➗ Splitting the Inequality: An absolute value inequality of the form $|ax + b| > c$ is equivalent to two separate inequalities: $ax + b > c$ or $ax + b < -c$.
• ➕ Solving the Inequalities: Solve each inequality separately for $x$. This typically involves algebraic manipulation, such as adding or subtracting constants and dividing by the coefficient of $x$.
• 🤝 Combining the Solutions: The solution to the original absolute value inequality is the union of the solutions to the two separate inequalities. This means that $x$ can satisfy either $ax + b > c$ or $ax + b < -c$.
• 📈 Graphical Interpretation: The solution can be visualized on a number line. The values of $x$ that satisfy the inequality will be those that lie outside the interval between the solutions of $ax + b = c$ and $ax + b = -c$.

🧮 Step-by-Step Solution

To solve an absolute value inequality of the form $|ax + b| > c$, follow these steps:

1. Split the Inequality: Create two separate inequalities: $ax + b > c$ and $ax + b < -c$.
2. Solve Each Inequality:
   • Solve $ax + b > c$ for $x$: Subtract $b$ from both sides to get $ax > c - b$, then divide by $a$ (remembering to flip the inequality sign if $a < 0$).
   • Solve $ax + b < -c$ for $x$: Subtract $b$ from both sides to get $ax < -c - b$, then divide by $a$ (again, flipping the inequality sign if $a < 0$).
3. Express the Solution: The solution is the union of the solutions to the two inequalities. Write the solution in interval notation or as a compound inequality.

🧪 Real-world Examples

Let's look at some examples:

1. Example 1: Solve $|2x - 1| > 5$.
   • Split: $2x - 1 > 5$ or $2x - 1 < -5$.
   • Solve:
     • $2x - 1 > 5 \Rightarrow 2x > 6 \Rightarrow x > 3$.
     • $2x - 1 < -5 \Rightarrow 2x < -4 \Rightarrow x < -2$.
   • Solution: $x > 3$ or $x < -2$. In interval notation: $(-\infty, -2) \cup (3, \infty)$.
2. Example 2: Solve $|-3x + 2| > 7$.
   • Split: $-3x + 2 > 7$ or $-3x + 2 < -7$.
   • Solve:
     • $-3x + 2 > 7 \Rightarrow -3x > 5 \Rightarrow x < -\frac{5}{3}$.
     • $-3x + 2 < -7 \Rightarrow -3x < -9 \Rightarrow x > 3$.
   • Solution: $x < -\frac{5}{3}$ or $x > 3$. In interval notation: $(-\infty, -\frac{5}{3}) \cup (3, \infty)$.

📊 Practice Quiz

Solve the following absolute value inequalities:

1. $|x + 4| > 2$
2. $|3x - 6| > 9$
3. $|-2x + 8| > 4$

Solutions:

1. $x < -6$ or $x > -2$
2. $x < -1$ or $x > 5$
3. $x < 2$ or $x > 6$

💡 Conclusion

Understanding and solving absolute value inequalities of the form $|ax + b| > c$ is a fundamental skill in algebra. By splitting the inequality into two separate cases and solving each one, you can find the set of all $x$ values that satisfy the original inequality. These concepts are important in many areas of mathematics and its applications.