Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the solution sets of the given inequalities. $$ |2 x-1|>2 $$

Step-by-Step Solution

Answer

The solution sets are \(x < -\frac{1}{2}\) or \(x > \frac{3}{2}\).

Step 1: Understand Absolute Inequalities

When dealing with an absolute value inequality, such as \(|a| > b\), where \(b > 0\), we can write two separate inequalities: 1. \(a > b\)2. \(a < -b\).This is because the absolute value expression \(|a|\) represents the distance from zero, so these two inequalities cover both the positive and negative directions.

Step 2: Identify the Expression Inside the Absolute Value

The expression inside the absolute value for our problem is \(2x - 1\). This will be used as \(a\) in our general inequality interpretation from Step 1.

Step 3: Write Two Inequalities

Based on the understanding of absolute inequalities, we set up two inequalities from the original problem \(|2x - 1| > 2\):1. \(2x - 1 > 2\)2. \(2x - 1 < -2\).

Step 4: Solve the First Inequality

Take the first inequality \(2x - 1 > 2\):- Add 1 to both sides: \(2x > 3\).- Divide both sides by 2: \(x > \frac{3}{2}\).

Step 5: Solve the Second Inequality

Now resolve the second inequality \(2x - 1 < -2\):- Add 1 to both sides: \(2x < -1\).- Divide both sides by 2: \(x < -\frac{1}{2}\).

Step 6: Combine Solution Sets

Combine the solutions from both inequalities, which are \(x > \frac{3}{2}\) and \(x < -\frac{1}{2}\). This means that any value of \(x\) either greater than \(\frac{3}{2}\) or less than \(-\frac{1}{2}\) satisfies the original absolute value inequality.

Key Concepts

solving inequalities

Solving inequalities can initially seem tricky, but it's like solving regular equations with only a couple of extra steps. An inequality expresses the relationship between quantities that are not equal, using symbols like greater than (\(>\)), less than (\(<\)), greater than or equal to (\(\geq\)), or less than or equal to (\(\leq\)). When you solve inequalities, your goal is to find the range or ranges of values that satisfy the inequality condition.

When working with inequalities, it's essential to remember a few rules:

• If you multiply or divide both sides of an inequality by a negative number, the inequality symbol flips direction. For example, \(-2x > 6\) becomes \(x < -3\) after dividing by \(-2\).
• The solution set may involve joining intervals, especially when dealing with absolute values. Explore both sides of the absolute condition to identify different ranges.

By carefully keeping these rules in mind, you can navigate through solving inequalities more efficiently and accurately.

algebraic manipulation

Algebraic manipulation plays a crucial role in solving inequalities, as it enables you to isolate the variable and determine the solution set. It involves performing various operations on both sides of an inequality. These operations include addition, subtraction, multiplication, or division, all done to simplify the equation and isolate the variable.

Let's consider the inequality \(2x - 1 > 2\). By algebraically manipulating the terms, we aim to get \(x\) by itself:

• Add 1 to both sides to remove the constant term on the left side, giving you \(2x > 3\).
• Divide each side by 2 to isolate \(x\), resulting in \(x > \frac{3}{2}\).

Similarly, for \(2x - 1 < -2\), we follow the same procedure:

• Add 1 to both sides: \(2x < -1\).
• Divide by 2: \(x < -\frac{1}{2}\).

Remember, the steps must be consistent and logical to ensure that the solution remains valid. By following these algebraic manipulations, you are one step closer to reaching the solution.

absolute value properties

Absolute value properties are essential for properly addressing absolute inequalities, as they establish how to interpret the equations. The absolute value of a number is its distance from zero on the number line, which means it's always non-negative. For a given expression \(|a| > b\) where \(b > 0\), the solution requires considering both possible directions from zero:

• \(a > b\)
• \(a < -b\)

In the problem of solving \(|2x - 1| > 2\), the property enables us to break it down into two separate inequalities: \(2x - 1 > 2\) and \(2x - 1 < -2\). These cover both potential scenarios:

• The distance being greater in the positive direction.
• The distance being greater in the negative direction.

Understanding these properties helps guide the algebraic steps needed to visualize both solution ranges. Absolute value properties are not just about simplifying equations but are fundamental to understanding and breaking down complex situations in mathematics.