Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |-x|>2 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -2) \cup (2, \infty)\).

Step 1: Analyze the Absolute Value Inequality

The given inequality is \(|-x| > 2\). Since the absolute value function \(|-x|\) is equivalent to \(|x|\), the inequality becomes \(|x| > 2\). An absolute value inequality of this form can be solved by considering two cases: when \(x > 2\) and when \(x < -2\).

Step 2: Solve Case 1 (x > 2)

For the first case, we are looking for all values of \(x\) such that \(x > 2\). This represents one part of the solution set.

Step 3: Solve Case 2 (x < -2)

For the second case, we need to find all values of \(x\) such that \(x < -2\). This represents the other part of the solution set.

Step 4: Combine the Solutions into an Interval

Combine the solutions from the two cases: \(x > 2\) or \(x < -2\). This represents two separate intervals, written as \((-\infty, -2) \cup (2, \infty)\).

Step 5: Verify the Solution

To ensure our solution is correct, pick test points from each interval, such as \(x = -3\) and \(x = 3\). Substituting these back into the original inequality \(|x| > 2\), both satisfy the inequality, confirming the solution set.

Key Concepts

Absolute Value Inequalities

When dealing with absolute value inequalities, it’s crucial to understand what the absolute value represents. The absolute value of a number is its distance from zero on the number line, regardless of direction. This means the absolute value is always non-negative. When solving inequalities like \(|x| > 2\), you are essentially looking for all the values of \(x\) that are more than 2 units away from zero. This results in two different scenarios because \(|x| > 2\) can be broken down into \(x > 2\) or \(x < -2\).
By transforming the initial inequality into these two conditions, the absolute value inequality becomes easier to solve. Each condition represents a separate part of the solution set. This basic understanding makes it much simpler to work through the problem.
Understanding absolute value inequalities is fundamental in algebra as they appear in various mathematical and real-world scenarios. From measuring uncertainties to calculating distances, mastering these inequalities offers a wide range of applications.

Interval Notation

Interval notation is a way of representing a set of numbers in a concise way, particularly useful for describing the solution of inequalities. With the inequality \(|x| > 2\), interval notation helps us express the answer neatly. The solution to this inequality is all numbers that satisfy \(x > 2\) or \(x < -2\).
In interval notation, these solutions are expressed as \((- \infty, -2) \) and \( (2, \infty)\). Together, they are written as \((- \infty, -2) \cup (2, \infty)\).
This notation uses:

• Parentheses \