Problem 27 - 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. \(x<0\)

Step-by-Step Solution

Answer

The subset represented by the inequality \(x<0\) includes all negative real numbers. It is represented on the real number line as an arrow pointing to the left, starting from but not including zero.

Step 1: Understand the inequality

The inequality, \(x<0\), describes all real numbers that are less than zero. This means any number that falls to the left of zero on the number line. It does not include zero, as indicated by the '<' symbol as opposed to '<=' which would include zero.

Step 2: Sketch the subset on the real number line

To sketch this subset on the real number line, first draw a straight horizontal line. This represents the number line. Mark a point in the middle of the line as zero. Draw an arrow pointing to the left starting from zero. The arrow represents all real numbers that are less than zero. Do not include zero in the arrow, you can illustrate that by making a small circle at the zero point, symbolizing that zero is not included in the set.

Step 3: Verbal description of the subset

The subset of real numbers that the inequality \(x<0\) represents includes all negative numbers but does not include zero. In simple terms, it's any number less than zero.

Key Concepts

Inequality

An inequality is a mathematical statement showing the relationship between two expressions that are not equal. In our exercise, the inequality is given as \(x < 0\), meaning we are interested in numbers less than zero. Inequalities like this are common in mathematics, and they help us describe ranges of numbers rather than specific numbers.

• "<" symbolizes "less than".
• \(x < 0\) indicates that zero is not part of the solution set.
• It describes an open-ended range extending infinitely in one direction on the number line.

Understanding inequalities is essential since they allow us to define segments of the number line where solutions lie. It is different from equations because equations state that two expressions are equivalent, while inequalities deal with ranges that are either greater or smaller.

Number Line

A number line is a visual tool used to help us understand the concept of real numbers more easily. It is a straight horizontal line on which we can represent numbers as points. Here's how you use a number line for inequalities like \(x < 0\):

• Begin by marking a point labeled as zero in the center of the line.
• Numbers to the left of the zero are negative, and numbers to the right are positive.
• To illustrate \(x < 0\), draw an arrow starting from zero and extending to the left.
• A small circle is often used at zero to show that zero isn't included in the set.

Utilizing a number line helps us to visually determine which numbers satisfy the inequality in question. This is especially helpful for students to grasp the concept of open-ended intervals and better understand the arrangement of numbers in relation to each other.

Negative Numbers

Negative numbers are an essential part of the real number system. They are the numbers that fall below zero on the number line. When working with the inequality \(x < 0\), you are essentially dealing with negative numbers.

• Negative numbers are the opposite of positive numbers and are typically preceded by a minus sign (e.g., -1, -2, -3).
• They increase in magnitude as we move to the left on the number line.
• Functions of negative numbers are prominent in various real-world situations, such as representing debts or temperatures below zero.

Understanding negative numbers is crucial because they frequently appear in mathematics and are necessary for solving problems involving inequalities like \(x < 0\). Negative numbers can be tricky at first, but with practice, much like working with any set of numbers, they will become more comfortable to use.