Question

graphing an inequality in two variables
graph ( y < \frac{1}{3}x + \frac{1}{3} )
click or tap the graph to plot a point.

Explanation:

Step1: Identify boundary line equation

The boundary is $y=\frac{1}{3}x+\frac{1}{2}$ (dashed, since $<$).

Step2: Find points for the line

Choose $x=0$: $y=\frac{1}{3}(0)+\frac{1}{2}=\frac{1}{2}$ → $(0, \frac{1}{2})$
Choose $x=3$: $y=\frac{1}{3}(3)+\frac{1}{2}=1+\frac{1}{2}=\frac{3}{2}$ → $(3, \frac{3}{2})$
Choose $x=-3$: $y=\frac{1}{3}(-3)+\frac{1}{2}=-1+\frac{1}{2}=-\frac{1}{2}$ → $(-3, -\frac{1}{2})$

Step3: Determine shaded region

Test $(0,0)$: $0 < \frac{1}{3}(0)+\frac{1}{2}$ → $0 < \frac{1}{2}$, which is true. Shade the region containing $(0,0)$.

Answer:

1. Plot the dashed line through points $(-3, -\frac{1}{2})$, $(0, \frac{1}{2})$, and $(3, \frac{3}{2})$.
2. Shade the region below this dashed line (the region that includes the origin $(0,0)$).
3. Table of points for the boundary line:

$x$	$y$
$0$	$\frac{1}{2}$
$3$	$\frac{3}{2}$