Problem 22 - Algebra 2 and Trigonometry

Question

In \(15-26,\) solve each inequality and write the solution set if the variable is an element of the set of integers. $$ |8+4 b| \geq 0 $$

Step-by-Step Solution

Answer

The solution set is all integers.

Step 1: Recognize Absolute Value Properties

Consider the properties of absolute values. An absolute value, such as \(|x|\), represents the distance from zero and is always non-negative. Therefore, \(|8 + 4b|\) is inherently greater than or equal to zero for any value of \(b\).

Step 2: Determine the Condition for Inequality

Since the inequality given is \(|8 + 4b| \geq 0\), this holds true for all possible values of \(b\), as the absolute value is always non-negative regardless of \(b\). Thus, there are no specific restrictions on \(b\) for this inequality.

Step 3: Define the Solution Set for Integers

Given that the inequality holds for all real numbers and \(b\) must be an integer, the solution set is all integers. This means every integer satisfies the inequality.

Key Concepts

Understanding Integers

In mathematics, integers include the set of whole numbers as well as their negative counterparts. More simply, this set is composed of numbers like

• -3, -2, -1 (the negative integers)
• 0 (neutral)
• 1, 2, 3, and so on (the positive integers)

These numbers do not include fractions or decimals. This is crucial because, when solving inequalities involving integers, we're only looking for whole-number solutions, without any fractional values.
In the context of our problem, any solution we find must be an integer. This means that if an inequality holds true for all real numbers, it automatically applies to all integers. This emphasizes the importance of knowing which subset of numbers (integers, in this case) we are dealing with to correctly interpret and express the solution.

Clarifying Absolute Value Properties

Absolute value refers to the non-negative value of a number without regard to its sign. For a number or expression inside the absolute value symbols, such as \(|x|\), it represents how far "x" is from zero on the number line, ignoring its direction. This always results in a value equal to or greater than zero.
This property is essential when dealing with inequalities involving absolute values. In the given inequality, \(|8 + 4b| \geq 0\), the inequality is automatically true for any expression inside the absolute value. The expression inside, \(8 + 4b\), could possibly be zero or any positive value. This implies that no matter what integer value "\(b\)" takes, the inequality \(|8 + 4b|\geq 0\) will remain satisfied. Understanding this property can simplify such inequalities greatly.

Decoding Inequalities

Inequalities express a range of possible values rather than just one exact solution. When an expression says \(x\geq0\), it essentially means that "x" can be zero or any positive value.
An inequality involving absolute values, such as \(|8 + 4b| \geq 0\), reveals that the expression can either be equal to zero or greater. This understanding helps clarify why the expression is true for any integer value of \(b\).
These inequalities don't restrict the values based on being positive or negative—they simply check whether the required condition (≥0, in this case) is met. Thus, such problems often have broad solution sets, which can include all real numbers—or, in our specific frame, all integers.

Defining the Solution Set

The solution set is a term used to describe all the values that satisfy a given inequality. When the task is to express the solution set just involving integers, the job is to find all whole numbers that make the inequality true.
In our problem, because \(|8 + 4b| \geq 0\) holds for all real numbers, it also automatically holds for all integers. Therefore, the solution set comprises all integers. There's no specific restriction or filtering necessary for any integer value.
Expressing the solution set in words, you would say it's the set of all integers. In essence, this means every integer, positive or negative, is a part of the solution set because each one satisfies the original inequality.