Problem 30 - Elementary Algebra

Question

Draw a number line that extends from -5 to \(5 .\) Is it possible to locate any numbers that are strictly greater than 3 but also strictly less than \(-2 ?\)

Step-by-Step Solution

Answer

Answer: No, there are no such numbers because the given intervals (3,∞) and (-∞,-2) do not overlap.

Step 1: Draw the number line from -5 to 5

To draw the number line, start by creating a straight horizontal line. Mark evenly spaced points on the line, labeling them as the integers from -5 to 5.

Step 2: Identify the intervals given in the problem

The problem gives us two intervals: (3,∞) and (-∞,-2). The interval (3,∞) includes all numbers strictly greater than 3, while the interval (-∞,-2) includes all numbers strictly less than -2.

Step 3: Visualize the intervals on the number line

On the number line, locate and circle the number 3. Similarly, locate and circle the number -2. The interval (3,∞) starts just after the circled 3 and extends to the right of the number line, while the interval (-∞,-2) starts just before the circled -2 and extends to the left of the number line.

Step 4: Check for overlaps between the two intervals

Notice that the two intervals do not overlap on the number line, as one is to the right of 3 and the other is to the left of -2. This means there are no numbers that are both in the interval (3,∞) and in the interval (-∞,-2).

Step 5: Answer the question

On the number line from -5 to 5, it is not possible to locate any numbers that are strictly greater than 3 but also strictly less than -2, because the given intervals do not overlap.

Key Concepts

Intervals

Intervals are an important concept in mathematics, especially when working with the real number line. They define a range of numbers between two specified points. There are different kinds of intervals:

• Open Intervals: Denoted by parentheses, such as \((a, b)\). They include numbers greater than \(a\) and less than \(b\), but not \(a\) or \(b\) themselves.
• Closed Intervals: Denoted by square brackets, such as \([a, b]\).These include all numbers between and including \(a\) and \(b\).
• Half-Open/Half-Closed Intervals: These intervals are a mix of open and closed, like \((a, b]\) or \([a, b)\), including one endpoint but not the other.

Interpreting intervals correctly allows us to understand which numbers belong in a set, crucial for solving inequalities.

Inequalities

Inequalities are used to express a relationship between two values where they are not equal. Inequalities like \(x > 3\) or \(x < -2\) involve comparing values on a number line. Here are some symbols used in inequalities:

• \(>\) (greater than) enables us to find values greater than a given number.
• \(<\) (less than) helps in locating values less than a specific point.
• \(\geq \) (greater than or equal to) and \(\leq \) (less than or equal to) extend the basic concept by including the number itself.

Understanding inequalities is crucial as they help define precise ranges of numbers that satisfy certain conditions. This is fundamental when plotting intervals, ensuring we accurately reflect parts of a number line.

Number Range

A number range refers to a span of numbers laid out across a number line. It is a visual representation of inequalities and intervals, helping identify which numbers are part of a specified set. When you draw a number line from \(-5\) to \(5\), you are creating a continuous range that includes all integers and real numbers within these boundaries.
A number range is not just limited to integers; it can represent all types of numbers, for instance negative numbers, fractions, and decimals, showcasing them in a linear order.
In the context of mathematical problems, such as determining if any numbers exist within both \((3, \infty)\) and\((-\infty, -2)\), the number range helps us visually and logically determine the solution. With the number line, it is evident that these intervals do not overlap, illustrating that there are no numbers simultaneously satisfying both conditions.