Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Appendices - Section A.1 - Real Numbers and the Real Line - Exercises A.1 - Page AP-6: 22


The equation is true for nonnegative numbers $\quad (a\geq 0),$ and is false for negative numbers $\quad (a\lt 0).$

Work Step by Step

The absolute value is defined as $\quad |x|=\left\{\begin{array}{ll} x, & x\geq 0\\ -x, & x \lt 0 \end{array}\right.$ Alternatively, as$\quad |x|=\sqrt{x^{2}.}$ Geometrically, on the number line, $|x|$ represents the distance (in units) from $x $ to $0.$ Note that $|x|$ can not be negative. So, if the RHS of the given equation is negative $\quad (a\lt 0)$, the equation can not be true. The equation stands true when RHS is nonnegative $\quad (a\geq 0)$.
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