Problem 17 - 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. $$ |b+6| \leq 5 $$

Step-by-Step Solution

Answer

The solution set is \{-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1\}.

Step 1: Understand the Absolute Value Inequality

The inequality \(|b+6| \leq 5\) means that the expression inside the absolute value, \(b+6\), is at most 5 units away from 0 on the number line. This sets up two inequalities: one for the positive scenario and another for the negative scenario.

Step 2: Solve the First Inequality

The first inequality from the absolute value statement is: \[b+6 \leq 5\]Subtract 6 from both sides to isolate \(b\):\[b \leq 5 - 6\]\[b \leq -1\]

Step 3: Solve the Second Inequality

The second inequality considers the negative scenario:\[b+6 \geq -5\]Subtract 6 from both sides to isolate \(b\):\[b \geq -5 - 6\]\[b \geq -11\]

Step 4: Combine Solutions

Combine the two inequalities to form a compound inequality:\[-11 \leq b \leq -1\]This describes the range of values that \(b\) can take, satisfying the original inequality.

Step 5: Determine the Integer Solution Set

Since \(b\) is an integer, list all integers between and including -11 and -1:The solution set is: \{-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1\}.

Key Concepts

Integer Solution Set

An integer solution set is a collection of integers that satisfies a given mathematical condition or inequality. In the exercise, the condition is given by the absolute value inequality \( |b+6| \leq 5 \). When we solve the inequality, we find that \(-11 \leq b \leq -1\).
The solution set only contains integers because the problem specifies that \( b\) must be an integer.
To form the integer solution set, list all possible integer values that fit within the range:

• Start from the smallest integer in the range, -11.
• Go up step by step through each integer up to the largest integer in the range, -1.

Thus, the integer solution set becomes: \{-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1\}.
This means any integer within this set will satisfy the original inequality.

Compound Inequality

A compound inequality is when two simple inequalities are combined into one statement. It provides a range of values that a variable can take. In our exercise, we formed the compound inequality \( -11 \leq b \leq -1 \) by solving \( |b+6| \leq 5 \).
Here's how the compound inequality was formed:

• The absolute value inequality was split into two distinct linear inequalities.
• We first solved for the scenario \(b+6 \leq 5\), which resulted in \(b \leq -1\).
• Then, we solved for the scenario \(b+6 \geq -5\), which gave us \(b \geq -11\).
• Combining these results, we obtained the compound inequality \(-11 \leq b \leq -1\).

This describes all the possible values that \(b\) can assume, satisfying both conditions at once.

Solve Inequalities

Solving inequalities involves finding all values that make the inequality true. When working with absolute value inequalities such as \(|b+6| \leq 5\), the process usually follows a set path:

• Understand the absolute value and set up two related inequalities.
• Handle each inequality separately, first representing non-negative scenarios, then negative scenarios.
• Simplify each inequality by solving them step-by-step, much like solving an equation.
• Combine the solutions to form a compound inequality that represents all possible solutions.

In our particular problem, by following these steps, we solved two linear inequalities and combined them to identify the range \(-11 \leq b \leq -1\).
This method ensures all parts of the inequality are dealt with, providing a comprehensive view of all possible solutions.