Problem 15 - Introductory Algebra for College Students

Question

Express the solution set of each inequality in interval notation and graph the interval. \(x>\frac{5}{2}\)

Step-by-Step Solution

Answer

The solution to the inequality \(x>\frac{5}{2}\) in interval notation is (\(\frac{5}{2}, \infty)\).

Step 1: Translate Inequality to Interval Notation

When \(x>\frac{5}{2}\), it means that x is any number greater than \(\frac{5}{2}\). In interval notation, this is expressed as \((\frac{5}{2}, \infty)\). The parenthesis ( is used instead of [ because x cannot be equal to \(\frac{5}{2}\).

Step 2: Create the Graph

The interval (\(\frac{5}{2}, \infty)\) can be graphed on a number line. Begin by marking \(\frac{5}{2}\) on the line. Because the interval does not include \(\frac{5}{2}\), a parenthesis ( or an open circle can be used to denote this. Then, draw an arrow to the right of \(\frac{5}{2}\), indicating that the interval continues towards positive infinity.

Key Concepts

Interval Notation

When dealing with inequalities, interval notation provides a clear and concise way to specify the set of all possible solutions. Simply put, interval notation uses a combination of parentheses \( ( ) \) and brackets \( [ ] \) to describe intervals on the real number line. For example, the inequality \( x > \frac{5}{2} \) can be translated into interval notation as \( (\frac{5}{2}, \infty) \).

Here's a quick guide to interpreting the symbols:

• \( (a, b) \): All numbers greater than \( a \) and less than \( b \), where neither \( a \) nor \( b \) are included in the solution set.
• \( [a, b] \): All numbers between \( a \) and \( b \) including \( a \) and \( b \).
• \( (a, \infty) \): All numbers greater than \( a \) with no upper limit (since infinity is not a number that can be reached).
• \( (\infty, b] \): All numbers less than \( b \) with no lower limit.

The symbols are chosen based on whether the endpoints are included in the solution set, indicated by brackets, or excluded, indicated by parentheses.

Number Line Graphing

Graphing on a number line is an essential skill in understanding inequalities. To graph the interval \( (\frac{5}{2}, \infty) \), you start by drawing a horizontal line, which represents a range of numbers extending from negative infinity to positive infinity. A specific value, such as \( \frac{5}{2} \), is marked on the line.

Since this value is not included in the solution, represented by the parenthesis in the interval notation, an open circle is placed at \( \frac{5}{2} \) on the number line. To show all the numbers greater than \( \frac{5}{2} \) that are included in the solution, an arrow is drawn extending rightward from the open circle towards infinity. This visual representation on the number line helps clarify which numbers satisfy the inequality.

Algebra Inequalities

Inequalities in algebra express the relationship between two expressions that are not necessarily equal to each other. Instead, one is either less than, greater than, less than or equal to, or greater than or equal to the other.

The inequality \( x > \frac{5}{2} \) illustrates that \( x \) can take any value greater than \( \frac{5}{2} \) but cannot be \( \frac{5}{2} \) itself. When solving inequalities, it is important to understand if the variable includes the boundary value (using \([ ]\)) or does not include it (using \(( )\)), impacting how the solution is expressed in interval notation and depicted on a number line graph.

Additionally, when solving algebraic inequalities, operations on the variable must maintain the inequality's direction, except when both sides are multiplied or divided by a negative number, in which case the inequality sign is reversed.