Problem 97 - Algebra and Trigonometry

Question

What's Wrong Here? It is tempting to try to solve an in- equality as if it were an equation. For instance, we might try to solve \(1<3 / x\) by multiplying both sides by \(x\) , to get \(x<3,\) so the solution would be \((-\infty, 3) .\) But that's wrong; for example, \(x=-1\) lies in this interval but does not satisfy the original in- equality. Explain why this method doesn't work (think about the sign of \(x\) ). Then solve the inequality correctly.

Step-by-Step Solution

Answer

The correct solution is \( x \in (0, 3) \).

Step 1: Analyzing the Mistake

In the inequality \( 1 < \frac{3}{x} \), multiplying both sides by \( x \) is problematic. This operation assumes \( x \) is positive. However, if \( x \) is negative, multiplying would reverse the inequality sign, leading to incorrect conclusions.

Step 2: Determine Sign of x

Consider scenarios where \( x \) is positive, negative, or zero. Specifically, values of \( x = 0 \) leave the expression undefined, and the behavior of \( x \) affects the direction of the inequality.

Step 3: Solve for Positive x

Assume \( x > 0 \), which maintains the direction of the inequality. Under this assumption, \( x \cdot 1 < 3 \cdot \frac{1}{x} \) simplifies to \( x < 3 \). Thus, the solution for positive \( x \) is \( 0 < x < 3 \).

Step 4: Solve for Negative x

Assume \( x < 0 \), which reverses the inequality sign. Multiplying by \( x \) leads to \( x > 3 \). However, since \( x < 0 \) inherently, \( x \) cannot be greater than any positive number, particularly \( 3 \). Therefore, there are no solutions when \( x < 0 \).

Step 5: Combine Results

Combine the solutions from both positive and negative cases. The correct solution is \( 0 < x < 3 \), as no valid solutions exist for \( x < 0 \). Exclude \( x = 0 \) since the original expression is undefined.

Key Concepts

Multiplication with Inequalities

Understanding inequalities is crucial because they behave differently than equations. When we multiply or divide both sides of an inequality by a number, the rules differ from equations.

One critical rule is that multiplying by a negative number reverses the inequality's direction. This is a key step that can trip up students. Unlike equations, inequalities require us to pay close attention to the sign of the number we use to multiply or divide.

• If you multiply both sides by a positive number, the inequality sign stays the same.
• If you multiply both sides by a negative number, you must flip the sign.

This rule ensures the inequality maintains its truth. Understanding why and how to apply this rule can lead to solving inequalities correctly.

Sign of Variables

When working with inequalities, the sign of the variables can change everything. In our example, the variable is \( x \), and its sign matters when we decide how to address the inequality \( 1 < \frac{3}{x} \).

Consider these scenarios:

• If \( x \) is positive, we can safely multiply or divide without flipping the inequality sign.
• If \( x \) is negative, multiplying or dividing both sides by \( x \) requires flipping the sign.

The sign also affects the bounds of our solution set. It's vital to examine each case separately. In our inequality, both positive and negative values of \( x \) lead to different solutions, or sometimes to no solution at all. Moreover, consider when \( x \) is zero, as this can make expressions undefined.

Direction of Inequality

The direction of an inequality tells us which side is greater or lesser. It's essential when performing operations like multiplication or division. If the inequality sign flips, it changes the entire solution.

Remember:

• Multiplying or dividing by a positive number keeps the inequality direction the same.
• Multiplying or dividing by a negative number flips it.

This flip can lead to incorrect solutions if not addressed or remembered. Practicing this will help you understand the mechanics and prevent errors when solving inequalities.

Undefined Expressions

Sometimes in mathematical expressions, certain values can lead to undefined results. In the inequality \( 1 < \frac{3}{x} \), if \( x = 0 \), this results in a division by zero, which is undefined. We must exclude such values from our solution set.

Why pay attention to undefined expressions?

• They highlight potential errors or gaps in expected solutions.
• Make sure those values are taken out of your solution set to maintain correctness.

Every time you solve an inequality involving fractions or divisions by variables, remember to check for values that might make the expression undefined. This step is crucial to finding the correct and valid solution.