Problem 69 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 1<|x-2|<4 $$

Step-by-Step Solution

Answer

The solution is \((-2, 1) \cup (3, 6)\).

Step 1: Understand the Absolute Value Inequality

The given inequality is composed of two parts: \(1 < |x-2|\) and \(|x-2| < 4\). Absolute value inequalities often translate into two inequalities. For \( |x-2| < 4 \), the equivalent is \(-4 < x-2 < 4\). Likewise, \(1 < |x-2|\) gives \(x-2 > 1\) or \(x-2 < -1\), considering both the positive and negative cases of the absolute value.

Step 2: Solve the Inequality \( -4 < x-2 < 4 \)

Add 2 to all parts of the compound inequality: \(-4 + 2 < x < 4 + 2\). This simplifies to \(-2 < x < 6\). This represents all values of \(x\) that satisfy \(|x-2| < 4\).

Step 3: Solve the Inequality \( x-2 > 1 \)

Add 2 on both sides of the inequality: \(x > 1 + 2\), which simplifies to \(x > 3\). This part represents all values of \(x\) that satisfy \(x - 2 > 1\).

Step 4: Solve the Inequality \( x-2 < -1 \)

Add 2 on both sides: \(x < -1 + 2\), simplifying to \(x < 1\). This part of the solution represents all \(x\) that satisfy \(x - 2 < -1\).

Step 5: Combine the Solutions

Combine the solutions, considering that the compound inequality \(-2 < x < 6\) overlaps with these intervals from steps 3 and 4. The overall solution is the union of intervals that satisfy both the main inequality parts: \(-2 < x < 1\) or \(3 < x < 6\).

Step 6: Express the Solution in Interval Notation

The solution in interval notation is \((-2, 1) \cup (3, 6)\). These intervals represent all the values of \(x\) satisfying the compound inequality 1 < |x-2| < 4.

Key Concepts

Compound Inequality

A compound inequality involves two separate inequalities that are connected by the words "and" or "or." This means that for a solution to be valid, it must satisfy the conditions of both inequalities. In our given exercise, the compound inequality is represented as \(1 < |x-2| < 4\). This particular format means we are looking for values of \(x\) that satisfy both conditions:

• \(1 < |x-2|\), implying the distance between \(x\) and 2 is more than 1.
• \(|x-2| < 4\), suggesting the distance is less than 4.

These intertwined requirements form a compound situation where the correct solution must meet each part simultaneously. By understanding this, one can find the intervals where \(x\) is the solution by solving each inequality separately and then looking for overlap or union as per the compound nature.

Absolute Value

Absolute value signifies the distance of a number from zero on the number line, without considering the sign. In the inequality \(|x-2|\), it measures how many units away \(x\) is from the number 2. For example, in \(1 < |x-2| < 4\), we are examining the range within which \(x\) holds a specific distance larger than 1 and smaller than 4 from 2.

• The inequality \(|x-2| < 4\) translates into \(-4 < x-2 < 4\), which is a classic form of an absolute value inequality.
• The inequality \(1 < |x-2|\) splits into \(x-2 > 1\) and \(x-2 < -1\), addressing both sides of the absolute value measurement.

This understanding helps simplify and solve the inequalities effectively by transforming them into linear forms without absolute values.

Interval Notation

Interval notation is a concise way to express a range of numbers or solutions. It uses brackets and parentheses to show which numbers are included or excluded in a set. In our problem, we express solutions using open intervals since the boundary values are not part of the solutions. For example:

• The solution \(-2 < x < 1\) is written as \((-2, 1)\) in interval notation, emphasizing exclusion of -2 and 1.
• Similarly, \(3 < x < 6\) is written as \((3, 6)\).
• We combine these using a union symbol \((\cup)\) to express: \((-2, 1) \cup (3, 6)\).

Interval notation offers a clear and efficient way to represent compound inequality solutions, especially useful in conveying solutions graphically or in calculations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations forming a mathematical phrase. In this exercise, expressions like \(x - 2\) help to define the inequalities.Exploring algebraic expressions involves several operations, such as:

• Addition and subtraction: Used to isolate \(x\) by moving constants to one side of the inequality, as in \(x - 2 > 1\), resulting in \(x > 3\).
• Multiplying or dividing, if necessary, which did not apply directly here but is often used in solving algebraic inequalities.

Understanding how to manipulate and solve these expressions is the foundation of solving more complex equations like those featuring absolute values.