Question

On the coordinates plane xOy hatch a set of points whose coordinates satisfy the following inequalities in two variables x and y (show the boundary belonging to the set by a solid line and that not belonging to the set by a dash line).
$lo{g}_{1/2}(x+y-1)>lo{g}_{1/2}y,\sqrt{y-x-1}<\sqrt{2-x}$

Answer 1

R.E.F image

Given inequality $lo{g}_{1/2}(x+y-1)>lo{g}_{1/2}y$

$\Rightarrow x+y-1<y...\left(1\right)$ $[\because lo{g}_{p}a>lo{g}_{p}b\Rightarrow a<bifp<1]$

Also, $x+y-1>0,y>\mathrm{0...}\left(2\right)$

inequality, $\sqrt{y-x-1}<\sqrt{2-x}\Rightarrow y-x-1<2-x$

$\Rightarrow y<\mathrm{3...}\left(3\right)$

Also, $y-x-1\ge 0,2-x\ge \mathrm{0...}\left(4\right)$

$\therefore$ Region is bounded by $x<1,y>0,x\le 2,y<3$

$x+y>1,y>x+1$

Test for origin

$1)x+y>1;$ Putting $(0,0)\Rightarrow 0>1$ [false]

does not contain origin

$2)y\ge x+1;$ Putting $(0,0)\Rightarrow 0>1$ [false]

does not contain origin