Question

GRAPHING Graph the system of linear inequalities. $$ \begin{aligned} &x+2 y<3\\\ &x-3 y>1 \end{aligned}$$

Short answer

The solution to the given system of inequalities is the overlapping area from shading below the line \(y=\frac{3-x}{2}\) and above the line \(y=\frac{x-1}{3}\).

Step-by-step solution

Convert inequalities

Convert the inequalities into 'y' form. The first one is \(x+2y<3\), subtract 'x' from both sides to get \(2y<3-x\), divide by 2 to get \(y<\frac{3-x}{2}\). Do the same with the second inequality to get \(y<\frac{x-1}{3}\).

Graphing inequalities

Start graphing these inequalities. First, plot the line of the first inequality \(y=\frac{3-x}{2}\). The slope is -1/2 and the y-intercept is 1.5. Next, plot the line for the second inequality \(y=\frac{x-1}{3}\). The slope is 1/3 and the y-intercept is -1/3. According to the inequalities, \(y<\frac{3-x}{2}\) and \(y>\frac{x-1}{3}\). This means for the first line we will shade below and for the second line we will shade above.

Identify solution area

The solution of the system of inequalities is where the shaded areas intersect. The intersection of the two shadings represents all the points (x, y) that satisfy both inequalities simultaneously.