Question

I need help with knowing which shaded region is the solution on the graph

Answer 1

Solution:

Given the system of inequalities:

$\begin{gathered} y<3x-5\text{ --- (1)} \\ y\ge-3x+2\text{ ----(2)} \end{gathered}$

From inequality (1),

$\begin{gathered} \text{when x=0, we have} \\ y<3(0)-5 \\ \Rightarrow y<-5 \\ \text{when y=0, we have} \\ 0\ge-3x+5 \\ \text{subtract 5 from both sides }of\text{ the inequality} \\ 0-5\ge3x+5-5 \\ -5>3x \\ divide\text{ both sides by the coeff}icient\text{ of x, which is 3.} \\ \text{thus,} \\ -\frac{5}{3}>\frac{3x}{3} \\ \Rightarrow-\frac{5}{3}>x \end{gathered}$

From inequality (2),

$\begin{gathered} \text{when x=0, we have} \\ y\ge-3(0)+2 \\ \Rightarrow y\ge2 \\ \text{when }y=0,\text{ we have} \\ 0\ge-3x+2 \\ \text{subtract }2\text{ from both sides of the inequality} \\ 0-2\ge-3x+2-2 \\ \Rightarrow-2\ge-3x \\ \text{divide both sides by the coefficient of x, which is -3.} \\ \text{thus,} \\ -\frac{2}{-3}\ge-\frac{3x}{-3} \\ \Rightarrow x\le\frac{2}{3} \\ \text{Note: when the coefficient of the unknown is negative, the inequality sign changes when dividing.} \end{gathered}$

Thus, plotting these range of values of x and y on a graph, we have

In the above plot, the intersections of the shaded regions of the inequalities is the solution to the system of inequalities.