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Chapter 1: Problem 116

Describe how to solve an absolute value inequality involving the symbol <. Give an example.

Short Answer

Expert verified
To solve an inequality involving '<', interpret it to mean that the value inside the absolute value sign must be less than the given number. Rewriting the inequality as a compound inequality will provide a solution set, expressed either as an inequality like \(-3 < x < 3\) or in interval notation as \((-3,3)\). These represent all real numbers between -3 and 3.

Step by step solution

01

Set Up the Inequality

The first step is to set up the inequality. For an inequality like \(|x| < a\), it represents that x is less than a units away from zero in both directions. As such, it can be rewritten as \(-a < x < a\). This structure will form the basis of solving the inequality.
02

Solve for x

In the case of \(|x| < a\), rewrite it as \(-a < x < a\), which means that x is between -a and a. Solve these two inequalities separately, which usually doesn't need further steps, as they basically give you your solution.
03

Write the Solution Set

The final step is to express the solution in set notation or interval notation. If a = 3, for example, the solution set will be \(-3 < x < 3\), which in interval notation is \((-3, 3)\). In this case, the solution indicates every real number between -3 and 3 is a solution to the inequality.

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Most popular questions from this chapter

In Exercises \(69-72,\) use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$ \frac{1}{(x-2)^{2}}>0 $$

Solve each quadratic inequality in Exercises \(1-28\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \left|x^{2}+2 x-36\right|>12 $$

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$5 \sqrt{-8}+3 \sqrt{-18}$$

In Exercises \(21-28,\) divide and express the result in standard form. $$\frac{5 i}{2-i}$$

In Exercises \(21-28,\) divide and express the result in standard form. $$\frac{2}{3-i}$$
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