Problem 8 - Introductory Algebra for College Students

Question

Graph the solutions of each inequality on a number line. \(x \leq 7.5\)

Step-by-Step Solution

Answer

The solution for the inequality \(x \leq 7.5\) includes 7.5 and all the numbers lower than 7.5. This is represented on a number line by a closed circle at 7.5 and a line extending to the left of this point.

Step 1: Identify the Inequality

Look at the inequality \(x \leq 7.5\). This inequality shows that x is less than or equal to 7.5.

Step 2: Draw the Number Line

Next step is to draw a number line. Look for where 7.5 would be if it were on this number line.

Step 3: Mark the Solution on the Number Line

As the inequality says that \(x\) is less than or equal to 7.5, make a closed circle at 7.5 to indicate that 7.5 itself is included in the solution. Then draw a line extending to the left side of the number line from 7.5 indicating all numbers less than 7.5.

Key Concepts

Number line

A number line is a visual representation of real numbers along a straight path. It helps in visualizing numerical values and inequalities effortlessly. You can think of it as a horizontal graph that shows numbers in order from left to right, increasing as they go to the right.
To create one, simply draw a straight line and mark even spaces for numbers. Each point on this line represents a real number. Numbers to the right are always greater than those to the left. Having a number line allows you to easily compare numbers and visualize expressions or inequalities.
Number lines are essential when dealing with inequalities as they provide a clear picture of which numbers are included in (or excluded from) a solution. This makes it extremely useful for answering questions or solving math problems.

Inequality solution

An inequality solution shows all the possible values that make the inequality true. In the example of the inequality \(x \leq 7.5\), it indicates all values of \(x\) that are less than or equal to 7.5.

Understanding the direction your inequality is going helps to know which numbers are part of the solution. For \(x \leq 7.5\), the solution consists of every number less than 7.5 plus the number 7.5 itself. Always understand the symbols:

• \(<\) or \(>\): The solution does not include the number itself.
• \(\leq\) or \(\geq\): The solution includes the number itself.

Visualize this on the number line by shading or marking the numbers that are solutions. This helps identify the range of the inequality at a glance.

Closed circle

A closed circle on a number line signals that a particular number is included in the solution of an inequality. This is crucial for inequalities using the symbols \(\leq\) or \(\geq\).

When you know your inequality includes the endpoint, place a solid, filled-in circle at that point. For instance, in \(x \leq 7.5\), you draw a closed circle at 7.5.
Closed circles are intuitive for understanding because they visually show that endpoints form part of the solutions. Compare this to an open circle, which indicates an endpoint is not included in the solution. Always remember:

• Closed circle: For \(\geq\) and \(\leq\)
• Open circle: For \(>\) and \(<\)

Having clear symbols like closed and open circles on a number line makes inequalities and their solutions easier to interpret and work with.