Basic Technical Mathematics, 11th Edition

17.4 Inequalities Involving Absolute Values

• Absolute Value Greater Than Given Value • Absolute Value Less Than Given Value

Inequalities involving absolute values are often useful in later topics in mathematics such as calculus and in applications such as the accuracy of measurements. In this section, we show the meaning of such inequalities and how they are solved.

If we wish to write the inequality $|\hspace{0.17em}x|\hspace{0.17em}\hspace{0.17em}>\hspace{0.17em}1$ without absolute-value signs, we must note that we are considering values of x that are numerically larger than 1. Thus, we may write this inequality in the equivalent form $x\hspace{0.17em}<\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}1$ or $x\hspace{0.17em}>\hspace{0.17em}1.$ We now note that the original inequality, with an absolute-value sign, can be written in terms of two equivalent inequalities, neither involving ...