Question

graphing an inequality in two variables\
graph: $y < \frac{1}{3}x + \frac{1}{2}$\
click or tap the graph to plot a point.

Explanation:

Step1: Identify boundary line equation

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

Step2: Find points for the line

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

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 below the line (where $(0,0)$ lies).

Table of Points:

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

Answer:

1. Plot the dashed line through points $(0, \frac{1}{2})$, $(3, \frac{3}{2})$, and $(-3, -\frac{1}{2})$.
2. Shade the entire region below this dashed line.