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Chapter 2: Problem 87

Solve each inequality. Write the solution set using interval notation. $$3-5|x|>-2$$

Short Answer

Expert verified
The solution set is \((-1, 1)\).

Step by step solution

01

Isolate the Absolute Value Expression

Subtract 3 from both sides of the inequality: \[3 - 5|x| - 3 > -2 - 3\]This simplifies to: \[-5|x| > -5\]
02

Divide by the Coefficient of the Absolute Value

Divide both sides of the inequality by \(-5\) and remember to reverse the inequality sign: \[|x| < 1\]
03

Create Compound Inequality

Rewrite the absolute value inequality as a compound inequality: \[-1 < x < 1\]
04

Express Solution in Interval Notation

Convert the compound inequality into interval notation: \((-1, 1)\)

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Key Concepts

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

Interval Notation
Interval notation is a way of writing subsets of the real number line. These intervals show which numbers are included (or excluded) in a solution set.
For example, the interval notation \((-1, 1)\) is read as 'all numbers between -1 and 1, but not including -1 and 1鈥�.
This is because parentheses ( ) indicate that the endpoints are not included, whereas brackets [ ] indicate that the endpoints are included.
In this exercise, the solution set is \((-1, 1)\). This means all numbers greater than -1 and less than 1 are part of the solution to the inequality.
It's essential to use the correct symbols to convey the right information:
Compound Inequality
A compound inequality consists of two inequalities that are combined into one statement by the words 'and' or 'or'.
These inequalities can be solved together and often express a range of values that satisfy both conditions simultaneously.
For the given exercise, the absolute value inequality \(|x| < 1\) is rewritten as a compound inequality: \(-1 < x < 1\).
This means x can be any number between -1 and 1 (not including -1 and 1).
The process includes breaking the absolute value into two separate inequalities and combining them.
Inequality Solving Steps
Solving inequalities often involves several important steps to ensure the correct solution:
  • Isolate the Absolute Value: Start by isolating the absolute value expression on one side of the inequality. In our exercise, that meant subtracting 3 from both sides to get \(-5|x| > -5\).
  • Divide by the Coefficient: Then, divide both sides by the coefficient of the absolute value term. Here, dividing by -5, and since we鈥檙e dividing by a negative, remember to reverse the inequality sign, resulting in \(|x| < 1\).
  • Create Compound Inequality: Rewrite the absolute value inequality into two separate inequalities that describe the same relationship. This step turns the inequality \(|x| < 1\) into \(-1 < x < 1\).
  • Express in Interval Notation: Finally, represent the solution set in interval notation, which is \((-1, 1)\).
By following these systematic steps, solving absolute value inequalities becomes a more straightforward task.

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