Problem 39 - Elementary Algebra

Question

Draw a number line that extends from -10 to \(10 .\) Place a point at all real numbers that are strictly greater than -8 but less than or equal to 7 .

Step-by-Step Solution

Answer

Answer: The range of real numbers that satisfy the given condition are -7 to 7, including 7 but not including -8. In interval notation, this can be represented as (-8, 7].

Step 1: Draw the Number Line

Draw a horizontal line and label the integers from -10 to 10 with equal spacing between each number.

Step 2: Identify the Range of Numbers

The condition given is that the numbers must be strictly greater than -8 but less than or equal to 7. This means that we need to mark all the real numbers larger than -8 (not including -8) and up to and including 7 on the number line.

Step 3: Place Points on the Number Line

Starting from the number -7 (since -8 is not included), place a point above the number line for each real number up to and including 7. To represent the continuous range of real numbers, draw a thick, solid line above the numbers from -7 to 7.

Step 4: Indicate the Inequalities

To show that -8 is not included in the range, draw an open circle at -8. For the endpoint at 7, which is included in the range, draw a closed circle. This way, it's clear that the range of real numbers starts just after -8 (without including it) and goes up to and includes 7.

Key Concepts

Inequalities

Inequalities are mathematical expressions that define a range or set of values that satisfy certain conditions. They are represented using symbols such as:

• <: less than
• >: greater than
• ≤: less than or equal to
• ≥: greater than or equal to

In the context of the original exercise, the inequality given is "strictly greater than -8 but less than or equal to 7." This means we need to consider numbers that fall within the range of being more than -8 and no larger than 7.
A solid line on a number line can represent continuous ranges of values allowed by the inequality. The use of open and closed circles denotes exclusion or inclusion of endpoints, respectively.
For example, an open circle at -8 indicates that -8 is not in the set (since numbers are *strictly* greater than -8), whereas a closed circle at 7 shows that 7 is included in the range (since numbers are *less than or equal to* 7). This graphical representation helps in visualizing inequalities clearly.

Real Numbers

Real numbers include all numbers that can be found on the number line. This encompasses:

• Whole numbers (0, 1, 2, 3, ...)
• Integers (..., -3, -2, -1, 0, 1, 2, ...)
• Rational numbers (such as fractions like 1/2, 2/3)
• Irrational numbers (such as √2, π)

Real numbers can be either positive or negative and include zero. They are the complete set of numbers that fill all possible points along the number line, making them crucial for representing continuous data sequences.
In practice, real numbers allow us to describe measurements and quantifiable data precisely. When marking real numbers on a graph, as in the exercise, remember that it includes all possible values between the endpoints, enabling a seamless continuum between points on the number line.

Interval Notation

Interval notation is a mathematical way of representing a range of values compactly using parentheses and brackets.

• Parentheses, like ( or ), indicate that an endpoint is not included (open interval).
• Brackets, like [ or ], indicate that an endpoint is included (closed interval).

For the inequality described in the exercise, which is "more than -8 but less than or equal to 7," the interval notation would be written as \((-8, 7]\).
This denotes that any value within this range is valid for the situation, except -8, which is excluded, while 7 is included.
Interval notation is useful because it offers a clear and concise way to express the same information that can also be represented on a number line or using inequalities, thus providing versatility in mathematical communication.