Question

Solve the following inequality:
$\sqrt{x-6}-\sqrt{10-x}⩾1.$

Answer 1

$\sqrt{(x-6)}-\sqrt{(10-x)}\ge 1$

Squaring both sides

$x-6+10-x-2\sqrt{x-6}\sqrt{10-x}\ge 1$

$4-1\ge 2\sqrt{x-6}\sqrt{10-x}\ge 1$

$3\ge 2\sqrt{x-6}\sqrt{10-x}$

$9\ge 4(x-6)(10-x)$

$9\ge 4[10x-60-x^2+6x]$

$9\ge 64x-240-4x^2$

$4x^2-64x+249\ge 0$

$\Rightarrow x=\frac{64\pm \sqrt{{64}^2-4\times 4\times 249}}{8}$

$=\frac{64\pm \sqrt{112}}{8}$

$\Rightarrow x=\frac{64\pm 4\sqrt{7}}{8}$

$=8\pm \frac{\sqrt{7}}{2}$

Hence, the answer is $8\pm \frac{\sqrt{7}}{2}.$