Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Give a verbal description of the subset of real numbers that is represented by the inequality, and sketch the subset on the real number line.\(-2

Step-by-Step Solution

Answer

The subset includes all real numbers strictly between -2 and 2. On the number line, it is represented by an open interval from -2 to 2, excluding -2 and 2.

Step 1: Understand the inequality

The inequality \(-2

Step 2: Verbal description

This subset includes all real numbers that are strictly between -2 and 2. It does not include -2 or 2 themselves because the inequality symbols '<' are used rather than '≤'.

Step 3: Sketch the subset on the real number line

On a real number line, this subset is represented by an open interval. Draw a line and mark points for -2 and 2. Draw parentheses facing outward at these points to signify that -2 and 2 are not included in the subset. The region between -2 and 2 represents all numbers in the subset.

Key Concepts

Real Numbers

Real numbers are a vast set of numbers that include both rational and irrational numbers. Rational numbers are those that can be expressed as a fraction or ratio of two integers, such as \(\frac{1}{2}\) or -3. Meanwhile, irrational numbers cannot be written as a simple fraction. Classic examples of irrational numbers include \(\pi\) or \(\sqrt{2}\), because they have non-repeating, non-ending decimal parts.
Real numbers cover everything found on the number line. This means all whole numbers, fractions, and decimals belong to this set. Whether you are dealing with positive, negative, or zero, they are considered real numbers.

• Negative Integer: -5
• Fraction: \(\frac{3}{4}\)
• Irrational Number: \(\sqrt{3}\)

In mathematical terms, real numbers are crucial because they help us perform calculations and plot them on a system we can visualize such as the number line.

Number Line

A number line is a visual tool used to represent real numbers in a simple, linear form. It is essentially a straight line with numbers placed at equal intervals along its length. The center of the number line is marked zero, with positive numbers stretching to the right and negative numbers extending to the left.
When sketching inequalities, a number line makes it easy to visualize which numbers are included and which are not. For example, the number line helps illustrate inequalities such as \(-2

The point at zero is central and acts as a reference point.

This line continues infinitely in both directions.

Each mark represents a specific, equidistant value unit.

Using a number line helps in understanding numerical relationships and making calculations easier.

Open Interval

An open interval describes a range of numbers between two endpoints, where the endpoints themselves are not included in the interval. Open intervals are typically denoted by using parentheses rather than square brackets.
For example, the open interval \( (a, b) \) includes all numbers between a and b, but not the numbers a and b themselves.
When dealing with inequalities like \( -2

Use round brackets to indicate open intervals (e.g., \( (-2, 2) \) ).

Draw parentheses on a number line at each endpoint to signify exclusion.

This approach is useful when describing continuous data, probabilities, and more.

Open intervals provide clarity in communicating which numbers fall within a certain range, allowing for a clearer understanding of data sets and constraints.

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Other exercises in this chapter

Problem 33

Simplify the expression.\(\frac{25 y^{8}}{10 y^{4}}\)

View solution

Problem 33

Identify the rule(s) of algebra illustrated by the statement.\(57 \cdot 1=57\)

View solution

Problem 33

Write the rational expression in simplest form.\(\frac{z^{3}-27}{z^{2}+3 z+9}\)

View solution

Problem 33

Factor the trinomial.\(3 x^{2}-5 x+2\)

View solution

Key Concepts

Real Numbers

Real numbers are a vast set of numbers that include both rational and irrational numbers. Rational numbers are those that can be expressed as a fraction or ratio of two integers, such as \(\frac{1}{2}\) or -3. Meanwhile, irrational numbers cannot be written as a simple fraction. Classic examples of irrational numbers include \(\pi\) or \(\sqrt{2}\), because they have non-repeating, non-ending decimal parts.
Real numbers cover everything found on the number line. This means all whole numbers, fractions, and decimals belong to this set. Whether you are dealing with positive, negative, or zero, they are considered real numbers.

• Negative Integer: -5
• Fraction: \(\frac{3}{4}\)
• Irrational Number: \(\sqrt{3}\)

In mathematical terms, real numbers are crucial because they help us perform calculations and plot them on a system we can visualize such as the number line.

Number Line

A number line is a visual tool used to represent real numbers in a simple, linear form. It is essentially a straight line with numbers placed at equal intervals along its length. The center of the number line is marked zero, with positive numbers stretching to the right and negative numbers extending to the left.
When sketching inequalities, a number line makes it easy to visualize which numbers are included and which are not. For example, the number line helps illustrate inequalities such as \(-2

The point at zero is central and acts as a reference point.

This line continues infinitely in both directions.

Each mark represents a specific, equidistant value unit.

Using a number line helps in understanding numerical relationships and making calculations easier.

Open Interval

An open interval describes a range of numbers between two endpoints, where the endpoints themselves are not included in the interval. Open intervals are typically denoted by using parentheses rather than square brackets.
For example, the open interval \( (a, b) \) includes all numbers between a and b, but not the numbers a and b themselves.
When dealing with inequalities like \( -2

Use round brackets to indicate open intervals (e.g., \( (-2, 2) \) ).

Draw parentheses on a number line at each endpoint to signify exclusion.

This approach is useful when describing continuous data, probabilities, and more.

Open intervals provide clarity in communicating which numbers fall within a certain range, allowing for a clearer understanding of data sets and constraints.