Problem 59 - Algebra for College Students

Question

Your friend looks at the inequality \(1+\frac{1}{x}>2\) and without any computation states that the solution set is all real numbers between 0 and 1 . How can she do that?

Step-by-Step Solution

Answer

The solution set is all real numbers between 0 and 1, i.e., \(0 < x < 1\).

Step 1: Rearrange the inequality

First, let's express the inequality in a more workable form. We start with the given inequality: \(1 + \frac{1}{x} > 2\). To isolate \(\frac{1}{x}\), subtract 1 from both sides: \(\frac{1}{x} > 1\).

Step 2: Solve for x

Next, we need to eliminate the fraction. When \(\frac{1}{x} > 1\), multiply both sides by \(x\). However, be cautious: the direction of the inequality sign will depend on whether \(x\) is positive or negative. This results in the inequality \(1 > x\), if \(x > 0\).

Step 3: Consider the solution domain

Since \(x\) must be positive to avoid division by zero, and because multiplying by a positive \(x\) retains the inequality direction, the domain where this holds true is \(0 < x < 1\). Therefore, the inequality is valid for positive \(x\) values less than 1.

Key Concepts

Solving Inequalities

Solving inequalities is a foundational skill in algebra that involves finding the set of values that satisfy a given inequality. Unlike equations, where the goal is to find specific values, inequalities often have several possible solutions.
When solving an inequality, such as \(1 + \frac{1}{x} > 2\), the process involves rearranging the inequality to isolate the variable in question. In this example, we subtract 1 from both sides to simplify the expression to \(\frac{1}{x} > 1\).

• Rearrange the inequality to make it easier to solve.
• Consider different cases for the variable value (e.g., positive, negative).
• Always verify the direction of the inequality after operations like multiplication or division.

Once rearranged, your attention shifts to finding values that satisfy this inequality. This often requires dividing by or multiplying both sides by a variable, as in our example where we multiply by \(x\) while keeping track of the inequality's direction. Multiplying by a positive number keeps the inequality sign unchanged, completing our transformation.

Inequality Properties

Understanding inequality properties is essential when manipulating inequalities. Unlike equations, inequalities are sensitive to operations involving multiplication or division by negative numbers. This is because multiplying or dividing both sides of an inequality by a negative number flips the inequality sign.
In our example, after transforming \(\frac{1}{x} > 1\), we multiply both sides by \(x\), noting that \(x\) must be positive. This requires special care since:\[ \text{If } x > 0, \; x \cdot \frac{1}{x} > x \cdot 1 \, \Rightarrow 1 > x. \]

• When multiplying or dividing by a variable, consider its sign to decide if the inequality direction changes.
• Addition and subtraction don't affect the inequality sign.

These rules and properties ensure we accurately determine the range of values that solve the inequality. Without these considerations, incorrect conclusions about the inequality's solution set might occur.

Solution Sets

A solution set is the collection of values that satisfy an inequality. For an inequality such as \(1 + \frac{1}{x} > 2\), identifying the correct solution set often involves understanding the range of possible values for \(x\).
In this problem, our friend zeroed in on the set \(0 < x < 1\) as the solution. This decision follows from understanding that:\[ \frac{1}{x} > 1 \implies 1 > x \text{ for } x > 0. \]

• The solution set is described by all real numbers that satisfy the conditions derived from solving the inequality.
• It can be represented on a number line or in interval notation.
• Always consider the domain restrictions, especially for divisions or roots.

Here, the solution set is \((0, 1)\), meaning \(x\) is any number greater than 0 and less than 1. This constraints consideration ensures accurate interpretation and representation of the solution set.