Problem 7 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The solutions to the inequality \(x \leq 4.5\) would be 4.5 and all numbers less than 4.5. On a number line, this would be represented with a filled circle at 4.5 and a line extending to the left, showing all the included values.

Step 1: Understand the inequality

The inequality \(x \leq 4.5\) means that 'x' could be 4.5 or any number less than 4.5.

Step 2: Draw a number line

Draw a number line and mark the number 4.5 clearly on it. The number line should have marks for whole numbers and also for fractions and decimal values close to 4.5.

Step 3: Highlight the solutions

Since the inequality includes values 'less than or equal to' four point five, the point representing \(4.5\) on the number line must be filled (usually represented by a circle filled with solid color). Then, draw a line extending to the left of 4.5, indicating that all numbers below 4.5 are also included in the solution.

Key Concepts

Number Line

A number line is a simple yet powerful tool to visualize numbers and their relationships. It is a straight line with numbers placed at equal intervals along its length. These numbers can include whole numbers, fractions, and decimals, making it versatile for all kinds of mathematical problems. In our case, we use a number line to show solutions to the inequality. It has numbers marked at equal distances, helping us see exactly where numbers fall in relation to each other.

• The center of the number line is usually zero, extending positively to the right and negatively to the left.
• Fractions and decimal points are crucial for illustrating values that are not whole numbers, such as 4.5.

Number lines are essential for understanding mathematical concepts at a glance and forming a visual idea of inequalities.

Graphing Inequalities

Graphing inequalities is a way to visually represent the range of solutions an inequality can have. The inequality we are working with is given by \(x \leq 4.5\).
To graph this:

• Start by marking the significant number, 4.5, on the number line.
• Since the inequality involves "less than or equal to," you fill in this point with a solid dot or circle. This tells us that 4.5 itself is included in the solution.
• Draw a continuous line extending to the left from the point 4.5. This line shows that all numbers less than 4.5 are also part of the solution.

The solid line and filled dot visually convey the entire set of solutions that satisfy the inequality as opposed to just one number or point.

Inequality Representation

Inequality representation uses symbols and diagrams to showcase relationships between quantities that aren't necessarily equal. Here, the inequality \(x \leq 4.5\) was used. Let's break it down:

• The symbol \(\leq\) indicates that 'x' can be equal to 4.5 or any number smaller. This gives us more than one possible solution.
• Graphically, inequalities are shown using lines and points on a number line. A filled circle or dot is used when the number is included in the solutions, as opposed to an open circle, which would denote exclusion.

Understanding inequality representation helps with recognizing the different possibilities that can satisfy an inequality rather than looking for a single specific answer. This form of expression is especially useful when analyzing data sets, mathematical functions, or problem-solving in real-world situations.