Problem 24 - 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. $$ |11-2 b|-6>11 $$

Step-by-Step Solution

Answer

The solution set is \( \{ b \in \mathbb{Z} \mid b < -3 \text{ or } b > 14 \} \).

Step 1: Simplify the Inequality

We start by isolating the absolute value on one side. Given the inequality \[ |11 - 2b| - 6 > 11, \]add 6 to both sides to eliminate it from the left:\[ |11 - 2b| > 17. \]

Step 2: Solve the Absolute Value Inequality

The absolute value inequality \( |11 - 2b| > 17 \) means there are two conditions to satisfy:1. \( 11 - 2b > 17 \)2. \( 11 - 2b < -17 \)

Step 3: Solve First Inequality Condition

For the first condition \( 11 - 2b > 17 \), subtract 11 from both sides:\[ -2b > 6. \]Next, divide both sides by -2, remembering to reverse the inequality sign:\[ b < -3. \]

Step 4: Solve Second Inequality Condition

For the second condition \( 11 - 2b < -17 \), subtract 11 from both sides:\[ -2b < -28. \]Then, divide both sides by -2, flipping the inequality sign:\[ b > 14. \]

Step 5: Combine Solution Conditions

Since the absolute value inequality required an 'or' condition, combine the solutions from Steps 3 and 4:\[ b < -3 \text{ or } b > 14. \]This means for the set of integers, \( b \) can take any integer value less than -3 or greater than 14.

Key Concepts

Integer Solution Sets

When dealing with inequalities where the variable belongs to the set of integers, we are looking for whole number solutions only. In the given inequality, we found that the solution for \( b \) is that it should be less than \(-3\) or greater than \(14\). This means,

• Integers like \(-4, -5, -6, \ldots\) are valid because they are less than \(-3\).
• Integers like \(15, 16, 17, \ldots\) are valid because they are greater than \(14\).

Therefore, the solution set consists of

• all negative integers starting from \(-4\) downward.
• all positive integers starting from \(15\) upwards.

This creates two distinct ranges where integers can fall, and it is important to realize these are separate, non-overlapping sets of numbers.

Inequality Solving Steps

To solve absolute value inequalities like \(|11 - 2b| - 6 > 11\), it's crucial to understand the step-by-step approach. Here's a simplified breakdown of solving such inequalities:

• Start by isolating the absolute value. You do this by performing operations that remove any constants positioned outside the absolute value expression. In this case, adding 6 to both sides yields the inequality \(|11 - 2b| > 17\).
• Next, break down the absolute value inequality into two separate inequalities. For \(|x| > a\), the two resulting inequalities are \(x > a\) and \(x < -a\). This dual condition arises because absolute value represents the distance from zero, needing to account for both positive and negative directions.
• Solve each inequality separately. Make sure to perform arithmetic operations carefully, paying attention to each step.

These steps help methodically break down the inequality, ensuring no step is overlooked.

Inequality Reversal with Division

One of the key aspects of solving inequalities is correctly handling operations that affect the inequality sign. This particularly applies when dividing or multiplying both sides by a negative number.

• Whenever you divide or multiply an inequality by a negative number, the direction of the inequality sign must be reversed.
• For instance, when solving \(-2b > 6\), dividing both sides by \(-2\) changes \(b\) to be less than \(-3\). So, \(b < -3\).
• Similarly, using \(-2b < -28\) required division by \(-2\), resulting in \(b > 14\).

Remembering to flip the inequality sign is crucial. Failing to do so will lead to incorrect solutions, which is why maintaining attention to detail in these steps can't be overlooked. Recognizing when the reversal occurs helps in achieving the correct solution set.