Problem 10 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The range for the variable \(x\) from the inequality \(-3 \leq x<6\) can be graphically presented on a number line as a shaded section from the point \(-3\) (inclusive) to the point \(6\) (exclusive), indicating that any number within this range is a solution to the inequality.

Step 1: Identify the inequality range

In the given inequality, \(-3 \leq x<6\), x is greater than or equal to \(-3\) and less than \(6\). This means all numbers from \(-3\) to \(6\) satisfy this inequality, including \(-3\), but excluding \(6\).

Step 2: Draw a number line

The number line will act as a visual representation of all the real numbers which can possibly be values for \(x\). Mark the points \(-3\) and \(6\) on this number line.

Step 3: Indicate the range of the inequality

Use a closed circle to represent \(-3\) on the number line, because the relation \(\leq\) includes the number \(-3\). However, use an open circle to represent \(6\) on the number line because the relation \(<\) does not include \(6\).

Step 4: Shade the defined range on the number line

Shade the section of the number line between (and including) \(-3\) up to, but not including \(6\). This represents all the values between \(-3\) and \(6\), where \(x\) is greater than or equal to \(-3\) and less than \(6\).

Key Concepts

Number Line

A number line is a visual tool that helps us understand numbers and their order. It's essentially like a ruler for numbers. Imagine a straight horizontal line with a series of numbers on it. This line can extend infinitely in both directions, but when we're solving problems, we focus on the relevant section of the line.
To effectively use a number line:

• Mark significant points, like our endpoints in an inequality, such as \(-3\) and \(6\).
• Place important numbers at even intervals for clear visualization.
• Always ensure the number line is properly labeled so the reference points are clear.

The number line is an excellent way to see where numbers lie in relation to one another, especially when dealing with inequalities.

Closed Circle

A closed circle on a number line indicates that the endpoint is included in the range of values. We use it when the inequality includes the endpoint, typically shown by the symbols \(\leq\) or \(\geq\).
In the inequality \-3 \leq x < 6\, a closed circle is placed at \-3\ because the values we consider include \-3\ as a valid solution. This closed circle tells us \-3\ is part of the solution set. It visually affirms that all the values starting from \-3\, but less than \(6\), satisfy the inequality.

Open Circle

An open circle is used on the number line to show that a particular endpoint is not included in the solution set. This usually corresponds with the inequality symbols \(>\) or \(<\).
In this exercise, \(-3 \leq x < 6\), the number \(6\) does not belong to the solution set because \(x\) is less than \(6\), not equal to it. Therefore, we place an open circle at \(6\) to show this exclusion. The open circle is a visual cue indicating the boundary of the set without including that point.

• Open circles highlight limits of a range without including the limit.
• It’s essential to correctly identify and depict these with every inequality to avoid errors.

Inequality Range

The inequality range is all the values that satisfy the conditions of an inequality. In the exercise given, \-3 \leq x < 6\, the range encompasses all numbers from \-3\ to just before \(6\).
To capture this:

• We start from \(-3\), using a closed circle because it is included in the range.
• We extend the coverage along the number line towards \(6\).
• At \(6\), use an open circle to indicate this number is not part of the solution.

This visual range helps you see all potential values that \(x\) might genuinely represent. Understanding the inequality range is crucial for correctly solving many math problems involving inequalities.