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Chapter 1: Problem 59

Solve each absolute value inequality. $$|x|<3$$

Short Answer

Expert verified
The solution to the inequality \(|x|<3\) is all x such that -3 < x < 3 or (-3,3) in interval notation.

Step by step solution

01

Establish the Inequality Range

An absolute value of a number is always a non-negative number. It is the distance of the number from zero on the number line. Here, \(|x|<3\) means that x is within a distance of 3 from 0. It can be interpreted as: -3 < x < 3
02

Divide Into Two Inequalities

We can break the inequality -3 < x < 3, into two separate inequalities and solve them separately. These are: Inequality 1: -3 < x Inequality 2: x < 3
03

Plotting the Solution

On a number line, plot the solutions for both inequalities. The answer is the overlap or intersection of the two inequalities.
04

Write Final Answer

The final answer is the set of all x such that -3 < x < 3. In interval notation, this can be written as (-3,3).

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