Question

for examples 6-9 solve and graph

1. 2x - 4y < 12

graphing inequalities in two variables

Explanation:

Step1: Isolate the $y$ term

$2x - 4y < 12$
$-4y < -2x + 12$

Step2: Solve for $y$ (reverse inequality)

$\frac{-4y}{-4} > \frac{-2x + 12}{-4}$
$y > \frac{1}{2}x - 3$

Step3: Identify boundary line

Boundary line: $y = \frac{1}{2}x - 3$ (dashed line, since inequality is $>$)

Step4: Find intercepts for graphing

• $x$-intercept: Set $y=0$

$0 = \frac{1}{2}x - 3$
$\frac{1}{2}x = 3$
$x = 6$ → Point $(6, 0)$

• $y$-intercept: Set $x=0$

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

Step5: Determine shaded region

Test point $(0,0)$: $0 > \frac{1}{2}(0) - 3$ → $0 > -3$ (true). Shade the region above the dashed line.

Answer:

The inequality simplifies to $\boldsymbol{y > \frac{1}{2}x - 3}$. To graph:

1. Draw a dashed line through $(6, 0)$ and $(0, -3)$ (the boundary $y=\frac{1}{2}x-3$).
2. Shade the region above the dashed line.