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 set is \(x \in (-\infty, 2) \cup (5, \infty)\).

Step 1: Understand the Absolute Value Inequality

The inequality \(|2x - 7| > 3\) tells us that the expression \(2x - 7\) is more than 3 units away from 0. This implies two separate conditions: \(2x - 7 > 3\) and \(2x - 7 < -3\).

Step 2: Solve the First Inequality

Start by solving \(2x - 7 > 3\). First, add 7 to both sides to isolate the term with \(x\): \(2x > 10\). Then divide both sides by 2 to solve for \(x\): \(x > 5\).

Step 3: Solve the Second Inequality

Now, solve the inequality \(2x - 7 < -3\). Add 7 to both sides: \(2x < 4\). Then divide both sides by 2 to isolate \(x\): \(x < 2\).

Step 4: Write the Solution Set

Combine the solutions from Step 2 and Step 3. These two solutions define the solution set for the original inequality: \(x > 5\) or \(x < 2\). Therefore, the solution set is \(x \in (-\infty, 2) \cup (5, \infty)\).

Key Concepts

Solving Inequalities

When it comes to absolute value inequalities like \(|2x - 7| > 3\), the goal is to find the range of values that satisfy the inequality. Absolute value inequalities often break down into two separate inequalities because the absolute value represents the distance from zero on a number line. So, in our example, the expression inside the absolute value \((2x - 7)\) must either be greater than 3 or less than -3.
The trick to handling these inequalities is to consider both possibilities.

• Condition 1: The expression inside can be greater than the positive threshold, i.e., \(2x - 7 > 3\).
• Condition 2: Alternatively, it can be less than the negative threshold, i.e., \(2x - 7 < -3\).

By solving each inequality separately, you determine the values of \(x\) that fulfill these conditions. Breaking it into smaller parts makes solving easier and clearer.

Solution Sets

The solution set of an inequality is essentially a collection of numbers that satisfy the given condition. With a compound inequality resulting from absolute values, you'll often find the need to express the solution using intervals.
When solving \(|2x - 7| > 3\), we derived two inequalities: \(x > 5\) and \(x < 2\). Each inequality provides a solution set, but together they describe the interval where the original inequality holds true.

• The first inequality, \(x > 5\), translates to an interval \((5, \infty)\).
• Meanwhile, the second inequality, \(x < 2\), corresponds to the interval \((-\infty, 2)\).

When you look at both conditions from the absolute inequality, they don't overlap. Therefore, they are combined in the form of a union: \((-\infty, 2) \cup (5, \infty)\). This union signifies that any real number in these intervals satisfies the original inequality.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a specific value or relationship. In the context of absolute value inequalities like \(|2x - 7| > 3\), understanding the structure of these expressions is crucial for solving them.
In the given exercise, the expression \(2x - 7\) is a linear expression. It combines two crucial operations:

• First, multiplying the variable \(x\) by 2.
• Second, subtracting 7 from the result.

When you solve the inequality, you manipulate this expression to isolate the variable. By understanding how expressions are constructed, you can confidently reorder and simplify them, whether that involves adding, subtracting, multiplying, or dividing. Recognizing these patterns will make working with inequalities more intuitive and less daunting.