Problem 23 - College Algebra

Question

Solve the inequality involving absolute value. Write your final answer in interval notation. $$ \left|\frac{x-3}{4}\right|<2 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-5, 11)\).

Step 1: Understand the Absolute Value Inequality

Absolute value inequalities of the form \(|A| < B\) can be rewritten as a compound inequality: \[-B < A < B\]This will help split the inequality to solve for \(x\) without the absolute value symbol.

Step 2: Set Up the Compound Inequality

For the inequality \(\left|\frac{x-3}{4}\right| < 2\), rewrite it as:\[-2 < \frac{x-3}{4} < 2\] This inequality stems from dealing with the absolute value and finding a range for the expression inside.

Step 3: Solve the Compound Inequality

To solve \(-2 < \frac{x-3}{4} < 2\), first, eliminate the fraction by multiplying the entire inequality by 4:\[-8 < x - 3 < 8\]Next, add 3 to all parts of the inequality to isolate \(x\):\[-8 + 3 < x < 8 + 3\]This simplifies to:\[-5 < x < 11\]

Step 4: Express the Solution in Interval Notation

The solution \(-5 < x < 11\) can be written in interval notation as \((-5, 11)\).This represents all numbers \(x\) that lie strictly between -5 and 11.

Key Concepts

Compound Inequalities

Compound inequalities involve two separate inequalities that are combined into one statement with either an 'and' or an 'or'. When dealing with absolute value inequalities, we often encounter the use of compound inequalities.

For instance, an absolute value inequality like \( |A| < B \) can be rewritten as a compound inequality:

• \(-B < A < B\)

This represents a range of values for the variable \(A\) that makes both parts of the compound inequality true at the same time. This 'and' condition is crucial.
Stepping through with an example: consider \( | rac{x-3}{4} | < 2 \). By removing the absolute value, we achieve the compound inequality:

• \(-2 < \frac{x-3}{4} < 2\)

This means values for \(x\) satisfy both \( \frac{x-3}{4} > -2 \) and \( \frac{x-3}{4} < 2 \) simultaneously, ultimately yielding a range of real numbers that solve the inequality.

Interval Notation

Interval notation is a way to describe sets of numbers within an interval, which can be a range where variables satisfy a certain condition.

The solution to a compound inequality, once solved, is often expressed in interval notation for simplicity and clarity. This involves using parentheses or brackets to show whether endpoints are included or excluded:

• Parentheses \(()\) indicate that the endpoint is not included (exclusive).
• Brackets \([]\) indicate that the endpoint is included (inclusive).

In our discussed example, we arrived at the inequality \(-5 < x < 11\). In interval notation, this solution is written as \((-5, 11)\), which conveys that \(x\) can take any value between \(-5\) and \(11\), but cannot equal \(-5\) or \(11\).

Interval notation helps streamline the communication of solution sets and is a universal tool in mathematics to describe ranges effectively.

Algebraic Expressions

Algebraic expressions are combination of variables, numbers, and operation symbols (+, -, *, /) that represents a mathematical relationship or rule.

In absolute value inequalities, these expressions often need to be manipulated to solve for a variable, which involves steps such as isolating the variable or clearing fractions. Consider the expression \( \frac{x-3}{4} \) in this exercise:

• Here, fractions are dealt with by multiplying through by the denominator, ensuring the inequality holds true during transformations.
• Once modified, algebraic operations help isolate \(x\), such as adding or subtracting terms from both sides of the inequality.

For example, multiplying the whole inequality \(-2 < \frac{x-3}{4} < 2\) by 4 results in \(-8 < x-3 < 8\). Further simplification involves adding 3 to all parts, finally leading to \(-5 < x < 11\).

Algebraic manipulation is a critical skill for efficiently finding solutions to equations and inequalities in mathematics.