Problem 33 - Algebra for College Students

Question

Graph the solution set for each compound inequality, and express the solution sets in interval notation. \(x>-1\) or \(x>2\)

Step-by-Step Solution

Answer

The solution in interval notation is \((2, \infty)\).

Step 1: Understand the Inequalities

The problem consists of two inequalities: \(x > -1\) and \(x > 2\). Both inequalities use the 'greater than' symbol.

Step 2: Interpret 'Or' in Inequality Context

The word 'or' in the compound inequality means that a number is part of the solution set if it satisfies at least one of the inequalities.

Step 3: Analyze Each Inequality Separately

For \(x > -1\), all numbers greater than -1 are included. For \(x > 2\), all numbers greater than 2 are included.

Step 4: Determine Overlapping Solution

The solution for \(x > 2\) includes all solutions of \(x > -1\) starting from 2. Thus, numbers greater than 2 satisfy both conditions, making the entire solution \(x > -1\) redundant.

Step 5: Express in Interval Notation

With \(x > 2\) being the more restrictive condition, the solution in interval notation is \((2, \, \infty)\).

Step 6: Graph the Solution Set

Graphically, the solution set is represented by a number line with an open circle at 2 and a line extending to the right towards infinity, indicating all numbers greater than 2.

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It is a compact form that uses parentheses and brackets to describe the range of numbers in a solution set. In the case of inequalities like "greater than" or "less than," parentheses are used to exclude endpoints. For example, for the inequality \(x > 2\), we use interval notation to write this as \((2, \infty)\). The parenthesis on 2 indicates that 2 is not part of the solution, while the infinity symbol always uses a parenthesis because infinity is a concept, not a specific number. Thus, \((2, \infty)\) represents all numbers greater than 2, extending infinitely.

Inequality Graphing

Graphing inequalities on a number line allows us to visually interpret the solution set. This helps in understanding which parts of the number line satisfy the inequality condition. For our example with \(x > 2\):

• Begin by drawing a number line.
• Identify the critical point, which is the number 2 in this case.
• Place an open circle on 2 to show that it is not included in the solution.
• Draw a line extending to the right from 2 to indicate that all numbers greater than 2 are included.

The open circle clearly indicates exclusion, while the arrow demonstrates extension into infinity. This method creates a simple and effective way to visualize where a solution set lies on the number line.

Solution Set Analysis

Solution set analysis involves understanding the range of values that are solutions to an inequality or a compound inequality, like in our exercise. This process includes the following steps:

• Considering each inequality individually to determine which values satisfy each condition.
• Recognizing how "or" compounds the inequalities, implying that satisfying just one is sufficient.
• Identifying the more restrictive condition; here it is \(x > 2\) because it is stricter than \(x > -1\).
• Presenting this refined solution set in a clean format like interval notation, \((2, \infty)\), ensuring clarity and precision.

By evaluating each part and concluding with the most informative solution set, you not only identify the correct solution but also deepen your understanding of how different solutions to inequalities relate to each other.