Problem 7 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph the solutions of each inequality on a number line. $$x \leq 4.5$$

Step-by-Step Solution

Answer

For the inequality \(x \leq 4.5\), the graph will have a filled circle at 4.5 (to represent '\(\leq\)') and an arrow pointing to the left (to represent all values of \(x\) that are less than 4.5).

Step 1: Recognize the type of inequality

Notice that the inequality symbol used is '\(\leq\)'. This means 'less than or equal to', which forms the basis of the understanding for this exercise.

Step 2: Identify the critical number

The number in the inequality that \(x\) is being compared to is 4.5. This is identified as the critical number.

Step 3: Draw the number line

The number line is then drawn and labelled. 4.5 is marked as the critical point on this number line.

Step 4: Graph the inequality

Since the inequality is \(x \leq 4.5\), the critical point 4.5 should be filled in (as it is included in the solution because of '\(\leq\)'), and an arrow pointing to the left of 4.5 should be drawn (since \(x\) is less than 4.5).

Key Concepts

Number Line

Understanding the concept of a number line is fundamental to graphing inequalities. A number line is a visual representation that showcases points corresponding to numbers placed in order along a straight line. Usually, the line is horizontal, with the left side indicating smaller values and the right side indicating larger values. Positive numbers are found to the right of zero, while negative numbers are to the left.

To start, draw a horizontal line and mark it with evenly spaced intervals. Labeling certain points can help provide context for where to plot the critical number. In the case of the exercise at hand, you would mark the critical point 4.5 on the number line to indicate where our inequality will be considered.

Inequality Symbols

Inequality symbols are mathematical notations representing the relationship between values. Key inequality symbols include '<' for 'less than', '>' for 'greater than', 'leq' for 'less than or equal to', and 'geq' for 'greater than or equal to'. They tell us not only the direction of the inequality but also whether the key value, the critical number, is included in the solution set.

In the exercise example with the inequality 'x leq 4.5', the 'leq' symbol indicates that the solution includes 4.5 itself (because it’s 'or equal to') and all numbers less than 4.5. Hence, the graph should have a filled circle or dot at 4.5 and a line or arrow extending to the left to indicate all the numbers that are less than 4.5.

Critical Number

The term 'critical number' refers to the specific value in an inequality that sets the boundary for the solution set. It is the value that the variable is being compared to, and in the context of graphing on a number line, it shows us where to mark the starting point of our inequality.

In the provided exercise, the critical number is 4.5. This number is 'critical' because it is the 'hinge' around which our inequality revolves; it tells us which numbers are included in the solution set based on the direction indicated by the inequality symbol. It's important to graph this number accurately—the entire solution graph depends on it. For 'x leq 4.5', this number will be represented on the number line with a solid dot to indicate it is part of the solution.