Problem 16 - Introductory and Intermediate Algebra for College Students 4th

Question

Express the solution set of each inequality in interval notation and graph the interval. $$x>\frac{7}{2}$$

Step-by-Step Solution

Answer

The solution set in interval notation is \( (\frac{7}{2}, \infty) \). The graph will show a line extending from the point \(\frac{7}{2}\) to a chosen large positive number, with an open circle at \(\frac{7}{2}\).

Step 1: Understanding Interval Notation

Interval notation is a way of writing subsets of the real number line. An interval is defined by its start point and its endpoint. For the inequality \(x > \frac{7}{2}\), the start point is \(\frac{7}{2}\) and the end point is not present because \(\frac{7}{2}\) is less than all other real numbers.

Step 2: Writing Interval Notation

In interval notation, the open interval \( (\frac{7}{2}, \infty) \) is used to represent all real numbers that are strictly greater than \(\frac{7}{2}\). The '(' symbol is used instead of the '[' symbol because \(x > \frac{7}{2}\), not \(x \geq \frac{7}{2}\). If it was \(x \geq \frac{7}{2}\), then '[' would be used.

Step 3: Drawing the Graph

On a number line, an open circle is used to represent the start of an open interval. For \(x > \frac{7}{2}\), an open circle should be drawn above the point on the number line that represents \(\frac{7}{2}\). Then, draw a line from \(\frac{7}{2}\) to some chosen large positive number (this represents \(\infty\)), indicating the numbers that are part of the solution.

Key Concepts

Interval Notation

When dealing with inequalities, interval notation is a concise way to express a range of values. It helps in representing all numbers between a given set of values on the real number line.
Here's how it works:

• An interval has start and end points, which could be numbers or infinity.
• Parentheses \( ( ) \) are used for open intervals, meaning endpoints are not inclusive.
• Brackets \[ [ ] \] indicate closed intervals, where endpoints are included.

In the inequality \(x > \frac{7}{2}\), the interval is written as \((\frac{7}{2}, \infty)\).
The use of the round bracket '(' indicates that \(\frac{7}{2}\) is not included in the solution.
Higher numbers are included, moving towards infinity.

Number Line

The number line is a visual representation that helps understand the order and relative position of numbers. It’s an essential tool for graphing inequalities.
To graph \(x > \frac{7}{2}\) on a number line:

• Identify the point corresponding to \(\frac{7}{2}\) on the line.
• Place an open circle at \(\frac{7}{2}\) to show it is not included in the solution.
• Draw a line extending to the right from the open circle, indicating all numbers greater than \(\frac{7}{2}\).

This graphical method clearly displays solutions and makes understanding inequalities more intuitive.

Algebra

Understanding algebra is crucial for solving and representing inequalities accurately. Algebra allows us to manipulate equations and inequalities to find the values that satisfy or solve the given condition.
In this case, \(x > \frac{7}{2}\) is an inequality. We aren't looking for one specific number, but all numbers greater than \(\frac{7}{2}\).
Steps to approach inequalities:

• Isolate the variable on one side.
• Manipulate the inequality just like an equation, adding or subtracting terms on both sides.
• Remember if you multiply or divide by a negative number, the inequality sign flips.

Mastery of these algebra concepts allows students to solve various inequalities, aiding in both academic and real-world problem-solving.