Problem 39 - Algebra A Combined Function

Question

Solve each inequality. Write the solution set in interval notation. $$ \frac{-1}{x-1}>-1 $$

Step-by-Step Solution

Answer

The solution set is \((0, 1) \cup (1, \infty)\).

Step 1: Understand the Inequality

We are given the inequality \( \frac{-1}{x-1} > -1 \). Our goal is to determine the values of \( x \) that satisfy this inequality.

Step 2: Simplify the Inequality

First, we need to isolate \( x \). We multiply both sides by \( -1 \) to remove the negative on the left. This also reverses the inequality sign: \(-\frac{1}{x-1} < 1 \).

Step 3: Solve the Inequality

Multiply both sides by \( x-1 \) to eliminate the fraction. This yields \(-1 < x-1\). Ensure to consider the possibility of \( x-1 \) being negative or positive, which can affect the inequality direction.

Step 4: Simplify Further

Add 1 to both sides of the inequality obtained in Step 3 to solve for \( x \): \(-1 + 1 < x \) or \(0 < x\). Hence, \( x > 0 \).

Step 5: Consider Restrictions and Domain

However, we must consider the original inequality \( x eq 1 \) because \( x-1 \) cannot be 0. Thus, the solution set must exclude \( x = 1 \).

Step 6: Write the Solution Set in Interval Notation

Considering \( x > 0 \) and \( x eq 1 \), the solution set is \((0, 1) \cup (1, \infty)\).

Key Concepts

Solution Set

When we solve an inequality, we are looking for all the values of the variable that make the inequality true. This collection of values is known as the "solution set". In simpler terms, the solution set is all the possible answers to the inequality. As we work through the solving process, we narrow down the potential values. In the case of the inequality \( \frac{-1}{x-1} > -1 \), we find that the solution set is

• \( x > 0 \): all positive numbers
• and \( x eq 1 \): excludes 1 because it makes the denominator zero

These conditions lead us to a solution set of \((0, 1) \cup (1, \infty)\). This notation indicates that \( x \) can be any number greater than 0, but not equal to 1.

Interval Notation

Interval notation is a way of writing the solution set for inequalities. It efficiently communicates which portions of the number line are included in the solution. Let's break down how interval notation works.For our example, the solution set is \((0, 1) \cup (1, \infty)\):

• \((0, 1)\) indicates that the values start just after 0 and go up to but do not include 1.
• \(\cup\) denotes the "union" of sets, meaning we combine them.
• \((1, \infty)\) means values start just after 1 continuing indefinitely to infinity.

Use round brackets \(( )\) to exclude end values, contrasting with square brackets \([ ]\), which include them. Thus, \( (0, 1) \) excludes 0 and 1, while \( (1, \infty) \) excludes 1 but includes values from slightly above 1 onwards.

Step-by-Step Solution

To solve inequalities, it's vital to follow a series of logical steps. Each step helps keep track of operations, ensuring the solution set remains accurate. Here’s a closer look.1. **Understand the Inequality:** Identify what needs to be solved. For example, the inequality \( \frac{-1}{x-1} > -1 \).
2. **Simplify the Inequality:** Manage signs and eliminate fractions by logically rearranging the inequality. Multiplying by negative values reverses inequality signs.
3. **Solve the Inequality:** Isolate the variable by performing algebraic operations like multiplication or addition on both sides.
4. **Consider Restrictions:** Verify constraints in the domain, such as avoiding division by zero or undefined operations.
5. **Write the Solution Set:** Express the valid values in interval notation, clearly showing which ranges are included.By breaking down inequalities into these manageable steps, one ensures a comprehensive approach that minimizes errors. This systematic process applied here led us to determine that \( x > 0 \) and \( x eq 1 \) yield the final solution \((0, 1) \cup (1, \infty)\).