Question

Derive the condition to be imposed on $\beta$ so that $(0,\beta )$ should be on or inside the triangle having sides
$y+3x+2=0,3y-2x-5=0$ and $4x+x-14=0$

Answer 1

The sides $BC,CA$ and $AB$ of the $△ABC$ are represented by equations
$4y+x-14=0$
$y+3x+2=0$
$3y-2x-5=0$ respectively as shown in the figure. From the figure, it is clear that if the point $(0,\beta )$ lies on or inside the $△ABC$, then the value of $\beta$ varies from $P(0,\frac{5}{3})$ to $Q(0,\frac{7}{2})$ that is $\frac{5}{3}\le \beta \frac{7}{2}$