Problem 42 - Elementary Algebra

Question

For the following problems, what numbers can replace \(x\) so that the following statements are true? \(-7

Step-by-Step Solution

Answer

Answer: The integer values of x that make the inequality true are -6, -5, -4, -3, and -2.

Step 1: Identify the range of integers

We are given the inequality \(-7 < x < -1\), which tells us that \(x\) must be greater than -7 and less than -1.

Step 2: List the integers between -7 and -1

To find the integers that meet this criterion, we can list the integers between -7 and -1. Remember not to include -7 and -1, as they are not part of the range. The integers are: \(-6, -5, -4, -3\), and \(-2\).

Step 3: Solution

The numbers that can replace \(x\) so that the statement is true are \(x = -6, -5, -4, -3\), and \(-2\).

Key Concepts

Integers

Integers are a fundamental part of mathematics. They include all the whole numbers, both positive and negative, as well as zero.
Some examples of integers are

• -5, -3, and 0
• 3 and 7
• 100 and -250

They don't include fractions, decimals, or numbers that have a fractional component.
When we are asked to identify the integers within a certain range, like in the original exercise, we only consider these whole numbers that fit the given criteria. In the original problem, we needed integers greater than -7 and less than -1. By listing them, we clearly see which numbers can be used as potential solutions.

Number Line

A number line is a helpful visual representation in mathematics that allows us to easily see the position of integers and other numbers.
This straightforward tool is essentially a straight line where every point represents a number, usually with zero placed in the middle. Negative integers extend to the left, while positive ones move to the right.
To visualize an inequality like equation \(-7 < x < -1\), the number line helps identify quickly which integers are included within this range.
Marking -7 and -1 with open circles indicates these numbers are not part of the solution set, while shading or highlighting the line segment between them includes only integer solutions like -6 through -2.
Using a number line can make the work of identifying possible integer solutions much simpler and more intuitive.

Inequality Notation

Inequality notation is an essential concept that indicates relationships between values. In mathematics, inequalities express how one value compares to another.
Common inequality symbols include

• \(<\), meaning "less than"
• \(>\), meaning "greater than"
• \(\leq\), meaning "less than or equal to"
• \(\geq\), meaning "greater than or equal to"

In the original exercise, we saw the use of the \(<\) symbol in \(-7 < x < -1\). This tells us that the integer \(x\) must be greater than -7 and less than -1.
Inequality notation is used to define and describe ranges quite efficiently, which is powerful especially when dealing with a set of numbers. Understanding and applying these concepts correctly helps solve equations and inequalities more accurately.