Question 1.36: Solve the following inequalities: (a) x²>4 (b) x²<4...
Solve the following inequalities:
(a) x²>4 (b) x²<4
For the product (x — 2)(x + 2) to be positive requires either
(i) x-2>0 \text { and } x+2>0
or
(ii) x-2<0 \text { and } x+2<0.
We examine each case in turn.
Case (i) \begin{aligned} & x-2>0 \text { and so } x>2 \\& x+2>0 \text { and so } x>-2\end{aligned}
Both of these are true only when x > 2.
Case (ii) \begin{aligned} & x-2<0 \text { and so } x<2 \\& x+2<0 \text { and so } x<-2\end{aligned}
Both of these are true only when x < —2. In summary, x² > 4 when x > 2 or x < —2.
\begin{aligned} (b)\quad x^2 & <4 \\ x^2-4 & <0 \\ (x-2)(x+2) & <0\end{aligned}For the product (x — 2)(x + 2) to be negative requires either
(i) x-2>0 \text { and } x+2<0
or
(ii) x-2<0 \text { and } x+2>0
We examine each case in turn.
Case (i) \begin{aligned} & x-2>0 \text { and so } x>2 \\ & x+2<0 \text { and so } x<-2 \end{aligned}
No values of x are possible.
Case (ii) \begin{aligned} & x-2<0 \text { and so } x<2 \\& x+2>0 \text { and so } x>-2\end{aligned}
Here we have x < 2 and x > —2. This is usually written as —2 < x < 2. Thus all values of x between —2 and 2 will ensure that x² < 4.
In summary, x² < 4 when —2 < x < 2.