Solving |ax + b| > c absolute value inequalities when c is negative

📚 Understanding Absolute Value Inequalities

Absolute value inequalities involve finding the range of values for a variable that satisfy a given inequality containing an absolute value expression. The absolute value of a number is its distance from zero. When we deal with inequalities of the form $|ax + b| > c$, where $c$ is a negative number, a unique situation arises.

📜 Historical Context

The concept of absolute value emerged from the need to define magnitude or size irrespective of direction or sign. While ancient mathematicians understood the idea of magnitude, the formalization of absolute value and its application in inequalities developed gradually. The notation and rigorous treatment of inequalities became more prominent in the 19th and 20th centuries.

🧠 Key Principle: Negative Values of c

When $c$ is negative, the inequality $|ax + b| > c$ is always true. This is because the absolute value of any expression is always non-negative (i.e., zero or positive). Therefore, any non-negative value will always be greater than a negative value.

📝 Proof

Let's consider the definition of absolute value: $|x| = egin{cases} x, & \text{if } x \ge 0 \\ -x, & \text{if } x < 0 \end{cases}$

Since $|ax + b|$ is always non-negative, it is always greater than any negative number $c$. Thus, the solution set is all real numbers.

✅ Solving |ax + b| > c when c is Negative: A Step-by-Step Guide

• 🔍 Identify the Inequality: Check if you have an inequality in the form $|ax + b| > c$.
• 💡 Check the Sign of 'c': Determine whether 'c' is a negative number.
• 📝 Conclude the Solution: If 'c' is negative, the solution is all real numbers. There's no further calculation needed!

🌍 Real-World Examples

Here are a few examples to illustrate this principle:

🔑 Key Takeaways

• 🌟 The absolute value of any expression is always non-negative.
• ➕ A non-negative value is always greater than a negative value.
• 🎯 Therefore, $|ax + b| > c$ is always true when $c$ is negative.

🎓 Conclusion

Solving absolute value inequalities where 'c' is negative becomes straightforward once you understand the fundamental principle that absolute values are always non-negative. When faced with $|ax + b| > c$ and $c < 0$, remember that the solution is simply all real numbers. No need for complex calculations—just a quick recognition of the negative value!