Question 1.1 • Calculus (4th Edition) • Textbookish

(i)

\begin{aligned} ax = a & \implies a^{-1} a x = a^{-1} a && a \neq 0 \\ & \implies x = 1 \end{aligned}

(ii)

\begin{aligned} x^{2} - y^{2} &= x^{2} - y^{2} + 0 \\ &= x^{2} - y^{2} + xy - xy \\ &= x (x + y) - y (x + y) \\ &= (x - y)(x + y) \end{aligned}

(iii)

\begin{aligned} x^{2} = y^{2} & \implies x^{2} - y^{2} = 0 \\ & \implies (x - y)(x + y) = 0 \\ & \implies x - y = 0 \; \vee \; x + y = 0 \\ & \implies x = y \; \vee \; x = -y \end{aligned}

Solutions