Problem 20 - 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. $$ |2 b-7| \geq 9 $$

Step-by-Step Solution

Answer

The solution set is \(\{b \in \mathbb{Z} \mid b \geq 8\} \cup \{b \in \mathbb{Z} \mid b \leq -1\}\).

Step 1: Understand the Absolute Value Inequality

The absolute value inequality \(|2b - 7| \geq 9\) means the expression inside the absolute value, \(2b - 7\), can be greater than or equal to 9 or less than or equal to -9.

Step 2: Separate into Two Inequalities

The inequality \(|2b - 7| \geq 9\) splits into two separate inequalities:1. \(2b - 7 \geq 9\)2. \(2b - 7 \leq -9\)

Step 3: Solve the First Inequality

Solve \(2b - 7 \geq 9\):1. Add 7 to both sides: \(2b \geq 16\).2. Divide by 2: \(b \geq 8\).

Step 4: Solve the Second Inequality

Solve \(2b - 7 \leq -9\):1. Add 7 to both sides: \(2b \leq -2\).2. Divide by 2: \(b \leq -1\).

Step 5: Combine the Solutions as a Set of Integers

The solution set combines the two solutions:1. \(b \geq 8\) or \(b \leq -1\),giving us two sets of integer solutions: \(\{b \in \mathbb{Z} \mid b \geq 8\} \cup \{b \in \mathbb{Z} \mid b \leq -1\}\).

Key Concepts

Integer Solutions

When dealing with inequalities, especially those containing absolute values, identifying integer solutions is crucial because they specify the set of discrete numbers that satisfy the inequality. In our exercise, we need to find integer solutions that satisfy the condition

• For the inequality \( |2b - 7| \geq 9 \), this means two separate integer ranges. These ranges are:
• \( b \geq 8 \), which translates to all integers from 8 to infinity.
• \( b \leq -1 \), giving us all integers from negative infinity up to -1.

The solution set is thus the combination of these two intervals. The integers found within these intervals meet the conditions laid out by the inequality, showcasing that such absolute value inequalities can have multiple sets of integer solutions.

Solving Inequalities

Solving inequalities involves finding the values of the variable that satisfy the inequality. With absolute value inequalities, the process starts by understanding that the function \( |x| \) identifies the non-negative distance of \( x \) from zero. Here's how you solve the inequalities:For an inequality like \(|2b - 7| \geq 9\):

• Start by setting up absolute value inequality into two separate inequalities: \(2b - 7 \geq 9\) and \(2b - 7 \leq -9\).
• For \( 2b - 7 \geq 9 \): Add 7 to both sides, leading to \( 2b \geq 16 \), then divide by 2 to find \( b \geq 8 \).
• For \( 2b - 7 \leq -9 \): Similarly, adding 7 gives \( 2b \leq -2 \), proceeding by dividing by 2 renders \( b \leq -1 \).

This splitting transforms the absolute value inequality into a classic expression, making it easier to find solutions.

Algebraic Expressions

Understanding algebraic expressions involves recognizing and manipulating expressions composed of variables, numbers, and operations. In the inequality \(|2b - 7| \geq 9\),the expression is centered around \(2b - 7\),a linear algebraic expression. Let's break it down:

• The terms within the expression are 2 multiplied by the variable \( b \),diminished by 7. This forms the heart of the absolute value inequality.
• These expressions become particularly critical when placed inside absolute value symbols. This dictates the two possible scenarios of solving the inequality:
• Either the expression yields a result greater than 9, or less than -9.

Recognizing these transformations aids in setting the stage for simpler interpretations of complex algebraic structures, such as the absolute value's capacity to branch into two potential inequalities.