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Magazine Chapter 2: Problem 69
Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 1<|x-2|<4 $$
Short Answer
Expert verified
The solution is \((-2, 1) \cup (3, 6)\).
Step by step solution
01
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.
02
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 \(|x-2| < 4\).
03
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 \(x - 2 > 1\).
04
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 \(x - 2 < -1\).
05
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 \(3 < x < 6\).
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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: \((-2, 1) \cup (3, 6)\).
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.
