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Chapter 2: Problem 116

Solve \(|2 x-9|<13 .\) Write the solution set in interval notation.

Short Answer

Expert verified
(-2, 11)

Step by step solution

01

- Remove the absolute value

The inequality \(|2x - 9| < 13\) can be broken down into two separate inequalities: \( 2x - 9 < 13\) and \(2x - 9 > -13\).
02

- Solve the first inequality

Solve \(2x - 9 < 13\): First, add 9 to both sides to isolate the term with the variable: \(2x < 22\).Then, divide both sides by 2: \(x < 11\).
03

- Solve the second inequality

Solve \(2x - 9 > -13\): First, add 9 to both sides to isolate the term with the variable: \(2x > -4\).Then, divide both sides by 2: \(x > -2\).
04

- Combine the two inequalities

The solution set is where both inequalities are true, which gives: \(-2 < x < 11\).
05

- Write the solution in interval notation

Convert the solution to interval notation: \((-2, 11)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Absolute Value
In mathematical terms, the absolute value of a number refers to its distance from zero on the number line, regardless of the direction.
It is always a non-negative value. The absolute value of a number \(a\) is written as \(|a|\).
For example, \(|-3| = 3\) and \(|5| = 5\).

The key idea behind absolute value inequalities, such as \(|2x - 9| < 13\), is that it describes a range within which the expression inside the absolute value must lie.
This inequality states that the expression \(2x - 9\) must lie between -13 and 13.
Solving Absolute Value Inequalities
To solve absolute value inequalities like \(|2x - 9| < 13\), you need to break it into two separate linear inequalities.
For \(|2x - 9| < 13\), it translates into:
  • \(2x - 9 < 13\)
  • \(2x - 9 > -13\)
This decomposition allows you to solve each part independently.
For the first inequality \(2x - 9 < 13\), solve by isolating x:
  • Add 9 to both sides: \(2x < 22\)
  • Divide by 2: \(x < 11\)
Similarly, for the second inequality \(2x - 9 > -13\):
  • Add 9 to both sides: \(2x > -4\)
  • Divide by 2: \(x > -2\)
This results in \(x\) values that satisfy both conditions simultaneously: \(-2 < x < 11\).
Interval Notation
Interval notation is a concise way of representing the solution set of inequalities.
It uses parentheses and brackets to denote the range of values.

Consider the solutions for \(-2 < x < 11\). This indicates all \(x\)-values between -2 and 11, but does not include -2 and 11 themselves.
In interval notation, this is written as \((-2, 11)\).

In general:
  • Parentheses \(( \, )\) are used when the endpoints are not included.
  • Brackets \([ \, ]\) are used when the endpoints are included.
This compact form is particularly useful in depicting solution sets for inequalities.
Always ensure it's clear whether the endpoints are included or not.

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