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Chapter 0: Problem 57

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

Short Answer

Expert verified
The solution to the absolute value inequality \(|x|<3\) is \(-3<x<3\).

Step by step solution

01

Understand the problem

We need to solve the absolute value inequality \(|x|<3\). This can be done by rewriting the inequality as two separate inequalities based off our knowledge of absolute value, namely that an absolute value measures distance from zero and thus is always positive and a number is less than the absolute value if it is within those 'bounds' on a number line.
02

Rewrite the absolute value inequality as two separate inequalities

Our inequality can be rewritten as \(x<3\) and \(-x<3\). We consider the second equation since the absolute value of a number is its distance from zero on a number line. Negative \(x\) represents the number's distance to the left of zero.
03

Solve the two separately

Solving \(x<3\) gives all numbers less than 3. For \(-x<3\), we multiply both sides by \(-1\) to remove the negative sign, but remember that multiplying or dividing an inequality by a negative number flips the sign of the inequality, so we get \(x>-3\).
04

Combine the solutions

Both conditions must be satisfied because absolute value means the number must be within the same distance of zero either positively and negatively. So, combining both solutions we get \(-3<x<3\).

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