Problem 2 - Algebra 2 and Trigonometry

Question

What is the solution set of \(\frac{|x|}{x}<0 ?\) Justify your answer.

Step-by-Step Solution

Answer

The solution set is \(x < 0\).

Step 1: Understanding the expression

The inequality given is \( \frac{|x|}{x}<0 \). This expression involves an absolute value function and division. The numerator \(|x|\) is the absolute value of \(x\), and the denominator is \(x\).

Step 2: Analyze the numerator

The numerator \(|x|\) represents the absolute value of \(x\). This is always non-negative, meaning \(|x| \geq 0\). The absolute value function results in 0 only when \(x = 0\).

Step 3: Analyze the fraction's inequality

For the fraction \( \frac{|x|}{x} \) to be less than zero, the fraction must be negative. Because \(|x|\) is non-negative, this implies that the denominator \(x\) must be negative to make the entire expression negative.

Step 4: Determine when the expression is undefined

The expression \( \frac{|x|}{x}\) is undefined at \(x = 0\), since you cannot divide by zero. Thus, \(x\) cannot be 0 in our solution set.

Step 5: Solution set conclusion

The fraction \( \frac{|x|}{x} \) is negative when \(x < 0\), so the solution set of the inequality is all negative real numbers.

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. So, when you see \(|x|\), it means you'll take the positive value of \(x\) regardless of whether \(x\) is originally positive or negative.
This makes absolute value always non-negative. The only time \(|x|\) equals zero is when \(x = 0\). However, be cautious! In our initial problem, \(|x|\) appears in a division with \(x\) as the denominator. Troubles arise here because division by zero is undefined. We'll touch on that in more detail later.

• The concept of absolute value ensures that the value is always greater than or equal to zero.
• It acts as a magnifier of a number's size, removing its direction.

Understanding absolute value is essential as it's directly linked to number manipulation in many algebraic expressions.

Working With Negative Numbers

Negative numbers are values less than zero on the number line. They represent the opposite of positive numbers. You encounter them often in real life such as when describing debts or temperatures below zero.
When you're solving inequalities like \(\frac{|x|}{x}<0\), negative numbers become central to finding a solution set. Here, the fraction becomes negative only if \(x\) itself is negative. This is because the numerator, \(|x|\), is always non-negative.

• Multiplying or dividing by a negative flips the inequality sign.
• Negative times positive results in a negative, which matches our inequality condition.

In mathematics, understanding how to manipulate and interpret negative numbers is key to solving equations and inequalities effectively.

Exploring Undefined Expressions

An expression is said to be undefined if its mathematical computation doesn't result in a real number. A common cause for undefined expressions is division by zero. When solving \(rac{|x|}{x}<0\), expression is undefined at \(x = 0\) because dividing by zero is not possible.
This occurrence is due to mathematical rules—division by zero doesn’t produce a meaningful result. So, any value of \(x\) that leads the denominator to be zero automatically makes the expression undefined and is excluded from the solution set.
Consider the caution required:

• Whenever solving inequalities, pay attention to when the expression becomes undefined.
• Avoid dividing by zero, and ensure no values in your solution set do this.

By understanding these precautions, you can avoid pitfalls that lead to incorrect mathematical reasoning.