202211041415 Solution to 1975-CE-AMATH-I-2

Find the values of $x$ satisfying both of the following inequalities:

$\begin{aligned} (1+x)(6-x)&\leqslant -8 \\ 3x-1 & \geqslant 5\\ \end{aligned}$

Roughwork.

The first inequality is

$\begin{aligned} (1+x)(6-x) & \leqslant -8 \\ 6+5x-x^2 \leqslant -8 \\ x^2-5x-14 \geqslant 0 \\ (x-7)(x+2) \geqslant 0 \\ x\leqslant -2\textrm{ \scriptsize{OR} }x\geqslant 7 \\ \end{aligned}$

whereas the second

$\begin{aligned} 3x-1 & \geqslant 5 \\ 3x & \geqslant 6 \\ x & \geqslant 2 \\ \end{aligned}$

In sum, $x\geqslant 7$ .

M	T	W	T	F	S	S
	1	2	3	4	5	6
7	8	9	10	11	12	13
14	15	16	17	18	19	20
21	22	23	24	25	26	27
28	29	30