Problem 1 - Intermediate Algebra

Question

Express the given inequality in interval notation and sketch a graph of the interval. \(x>1\)

Step-by-Step Solution

Answer

The interval is \((1, \infty)\), starting just above 1 and extending to infinity.

Step 1: Understanding the inequality

We start with the inequality \(x > 1\). This inequality indicates that we are looking for all values of \(x\) that are greater than 1.

Step 2: Identifying the interval

Since \(x > 1\) means all numbers greater than 1 (but not including 1 itself), the interval is expressed as \((1, \, \infty)\). This notation indicates all numbers from 1 to infinity, excluding 1.

Step 3: Sketching the graph

On a number line, you'll draw a circle at 1 to indicate that it is not included in the solution set. Then, shade or draw an arrow to the right of 1 to show all numbers greater than 1, extending indefinitely.

Key Concepts

Inequalities

Inequalities are mathematical expressions that compare two values. They express a relationship where one value is either greater than, less than, greater than or equal to, or less than or equal to another value. The inequality symbol signifies this relationship. Here are some common symbols used in inequalities:

• ">" : greater than
• "<" : less than
• "≥" : greater than or equal to
• "≤" : less than or equal to

In our example, the inequality is expressed as \(x > 1\). This indicates that the possible values of \(x\) are greater than 1. It does not, however, include the number 1 itself in the set of values that \(x\) can take. Inequalities are essential in defining ranges of values that satisfy certain conditions, which is why understanding them is critical.

Number Line Graph

A number line graph is a visual tool used to represent numbers and intervals. It's a straightforward way to visualize an inequality, making the comprehension of solutions easier. When we graph inequalities such as \(x > 1\), it provides a clear image of which numbers belong to the solution set.
To graph the inequality \(x > 1\), we:

• Draw a horizontal line to represent the number line.
• Mark the point where \(x = 1\) on the line.
• Use an open circle to indicate that 1 is not included in the solution.
• Shade or draw an arrow extending to the right, illustrating that \(x\) can be any number greater than 1.

This graph effectively communicates that all numbers greater than 1 are part of the solution, visually reiterating the concept the inequality conveys.

Solution Set

The solution set of an inequality is the collection of all possible values that satisfy the inequality. In mathematical terms, it represents the values of a variable that make the inequality true. For \(x > 1\), the solution set includes all numbers greater than 1.
When expressing the solution set of an inequality, interval notation is often used. This notation is concise and efficient for indicating the range of values. For example, the solution set for \(x > 1\) is written as \((1, \infty)\), where:

• "\((\)" indicates that the endpoint is not included.
• The number "1" is the lower boundary of the interval.
• "\(\infty\)" (infinity) signifies that the interval extends indefinitely.

Interval notation is beneficial as it provides a standard format to represent both finite and infinite ranges, making it easier to read and understand the solution sets of inequalities.