Problem 1 - Introductory Algebra for College Students

Question

Graph the solutions of each inequality on a number line. \(x>5\)

Step-by-Step Solution

Answer

The graphed solution for the inequality \(x > 5\) represents all the numbers to the right of 5 on the number line.

Step 1: Understand the inequality

The inequality \(x > 5\) means that \(x\) is any number greater than 5.

Step 2: Draw a number line

Draw a number line. Put a few numbers on it to provide a frame of reference. Be sure to include the number 5, as well as some numbers larger than 5.

Step 3: Identify the number

On the number line, identify the number 5 which is the boundary for this inequality.

Step 4: Graph the solution

Since \(x\) is greater than 5, all numbers to the right of 5 on the number line are part of the solution. Draw an open circle at 5 as 5 is not included in the solutions and an arrow towards the right to indicate that every number greater than 5 is part of solution.

Key Concepts

Number Line

A number line is a visual representation of numbers placed in order along a straight horizontal line. It helps to easily compare and understand the positions and relationships between different numbers. When solving inequalities, a number line is an essential tool.

To construct a number line:

• Draw a straight, horizontal line.
• Place evenly spaced tick marks along the line.
• Label these tick marks with integers such as 0, 1, 2, 3, etc., to serve as reference points.

It's important to include any crucial points relevant to the inequality you are working with. In the exercise, we include the number 5, since it's the boundary of our inequality.

Inequalities

Inequalities represent a range of values that satisfy a given condition. Instead of equality, which denotes an exact value, an inequality indicates that a variable can be greater than, less than, or fall within a set of values.

There are several symbols used in inequalities:

• ">" means greater than.
• "<" means less than.
• ">=" means greater than or equal to.
• "<=" means less than or equal to.

For the original exercise, we dealt with the inequality \(x > 5\). This means any value of \(x\) must be more than 5 but cannot include 5 itself. Therefore, the open circle at 5 signifies that 5 is excluded from the solution set.

Solution Set

The solution set of an inequality consists of all the values that satisfy the inequality condition. This is the collection of numbers that makes the inequality true when substituted for the variable in question.

For the inequality \(x > 5\), the solution set is all real numbers greater than 5. When graphing on a number line, we use:

• An open circle to represent that a number is not included in the solution set.
• An arrow or line extending in the appropriate direction to show all the numbers that are part of the solution set. For \(x > 5\), the arrow points to the right.

This visual cue helps see at a glance which values are solutions, providing a quick understanding of the inequality's range.