Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the solution sets of the given inequalities. $$ |2 x-7|>3 $$

Step-by-Step Solution

Answer

The solution sets are \(x < 2\) or \(x > 5\).

Step 1: Understand the Absolute Value Inequality

The inequality \(|2x - 7| > 3\) means that the expression inside the absolute value, \(2x - 7\), is either greater than 3 or less than -3. We will split this inequality into two separate inequalities to find the solution sets.

Step 2: Split into Two Inequalities

Given the original inequality \(|2x - 7| > 3\), we have two cases: 1. \(2x - 7 > 3\)2. \(2x - 7 < -3\). We will solve each case separately.

Step 3: Solve the First Inequality

Solve \(2x - 7 > 3\).- Add 7 to both sides to get: \(2x > 10\).- Divide each side by 2: \(x > 5\).

Step 4: Solve the Second Inequality

Solve \(2x - 7 < -3\).- Add 7 to both sides to get: \(2x < 4\).- Divide each side by 2: \(x < 2\).

Step 5: Combine the Solution Sets

The solution to \(|2x - 7| > 3\) is the combination of the solutions from Steps 3 and 4.- So, the solution sets are \(x < 2\) or \(x > 5\). These solutions are not connected and are separate ranges of allowed values for \(x\).

Key Concepts

Solution Sets in Absolute Value Inequalities

When we talk about solution sets in the context of absolute value inequalities like \(|2x - 7| > 3\), we're referring to the range or ranges of values that satisfy the inequality. In simple terms, the solution set is the collection of all possible values of \(x\) that make the inequality true.

This means we are looking for values of \(x\) that either make \(2x - 7\) greater than 3 or less than -3. These solution sets are determined by solving the inequalities that arise after removing the absolute value.

The solution sets help us understand where our solution lies on the number line, often resulting in ranges rather than specific numbers. In our case, solving \(|2x - 7| > 3\) gives us the solution sets \(x < 2\) or \(x > 5\), showing us that \(x\) can take on a wide array of values outside the interval [2, 5].

Splitting Inequalities In Absolute Values

Splitting inequalities is a crucial step when dealing with absolute value inequalities. Absolute values can hide certain features of an equation or inequality because they make everything positive. Therefore, splitting helps us uncover the conditions "hidden" by the absolute value signs.

For an absolute value inequality like \(|2x - 7| > 3\), this method involves separating the inequality into two distinct inequalities. Why two? Because absolute values measure the distance from zero on both the positive and negative sides.

Here's how it works for our problem:

• First, consider the positive scenario: \(2x - 7 > 3\).
• Then, consider the negative scenario: \(2x - 7 < -3\).

Both inequalities need to be solved separately. Each of these gives a part of the solution set; their union embodies the full range of solutions possible where \(|2x - 7|\) is greater than 3.

Steps to Solve Inequalities with Absolute Values

Solving inequalities with absolute values is a systematic process that allows us to find the solution set effectively. Let's break down the steps using the inequality \(|2x - 7| > 3\) as an example:

Step 1: Understand the Inequality
Recognize that the absolute value means the expression could be greater than the value OR less than the negative of that value.

Step 2: Split the Inequality
Divide into two separate inequalities:

• \(2x - 7 > 3\)
• \(2x - 7 < -3\)

Step 3: Solve Each Inequality Separately
For each inequality, solve for \(x\):

• Add or subtract to isolate terms containing \(x\).
• Divide to solve for \(x\).
• Ensure you flip the inequality sign when multiplying/dividing by a negative.

Step 4: Combine Solutions
The solution set will be the union of the solutions from the two inequalities. In this case, \(x < 2\) or \(x > 5\).

By following these steps, you ensure that you have rigorously derived all possible values of \(x\) that satisfy the original inequality.