Problem 10 - Introductory Algebra for College Students

Question

Graph the solutions of each inequality on a number line. \(-3 \leq x<6\)

Step-by-Step Solution

Answer

The graph of the solutions to the inequality \(-3 \leq x < 6\) is represented on a number line with a closed circle at -3, an open circle at 6, and a line connecting them.

Step 1: Identify the bounds of the interval

The inequality \(-3 \leq x < 6\) specifies a range for \(x\). So the values of \(x\) we are concerned with are from -3 to 6 on the number line.

Step 2: Represent the lower bound

First, we need to represent the lower bound of the interval on the number line. Since \(x\) is greater than or equal to -3 (\(-3 \leq x\)), we represent this with a closed circle (indicating that the number -3 is included in the solutions) at -3 on the number line.

Step 3: Represent the upper bound

Next, we represent the higher bound of the interval on the number line. Since \(x\) is less than 6 (\(x < 6\)), we signify this with a open circle (indicating that the number 6 is not included in the solutions) at 6 on the number line.

Step 4: Graph the interval

Now, draw a line connecting the closed circle at -3 and the open circle at 6. This line represents all possible values of \(x\) that satisfy the inequality, with every point on the line being a solution to the inequality \(-3 \leq x < 6\).

Key Concepts

Number Line

A number line is a simple visual tool that helps us represent numbers in a linear format. Imagine it as a horizontal line with numbers placed at regular intervals. Numbers increase from left to right, with zero typically in the center. A number line is not just for integers; it can also include fractions and decimals. For inequalities like \(-3 \leq x < 6\), a number line becomes especially useful. It shows the range of values that a variable, in this case, \(x\), can take. When working with inequalities, number lines help us

• Visualize the inequality by marking boundaries.
• Identify points of inclusion or exclusion with symbols like closed or open circles.
• Spot easily where the solutions for the inequality lie along the line.

This makes number lines an essential tool in understanding and solving inequalities.

Closed Circle

A closed circle on a number line is used to show that a particular number is included in the range of solutions to an inequality. Consider the inequality \(-3 \leq x < 6\). The symbol \(\leq\) tells us that \(-3\) is a part of the solution. Thus, we represent \(-3\) on the number line with a closed circle. To create a closed circle:

• Find the number on the number line, like \(-3\).
• Draw a solid circle around that point.

This indicates that the number is included. Closed circles are "filled" indicating that the boundary value is part of the set of solutions. This simple mark makes it visually clear which boundaries are included in the inequality solution.

Open Circle

An open circle on a number line is used to show that a number is not included in the range of solutions to an inequality. In the example of \(-3 \leq x < 6\), the less-than symbol \(<\) with \(6\) signifies exclusion. Thus, \(6\) is marked with an open circle. To illustrate this:

• Locate the number, such as \(6\), on the number line.
• Draw a circle around this point but keep it empty.

This empty circle suggests that the number itself is not part of the solution set. Open circles help clarify which values are not included, making it easy to understand the limitations of the inequality.