Question

question 14 (essay worth 10 points)
(graphing systems of inequalities hc)
the following system of inequalities is given
$x + 2y \geq 16$
$x + y < 20$
part a: explain how to graph the solution for the system of inequalities. be sure to include the type of line and the direction of the shading. (6 points)
part b: determine if the ordered pair $(-10, 20)$ is in the solution set. explain why or why not. (2 points)
part c: determine if the ordered pair $(20, 10)$ is in the solution set. explain why or why not. (2 points)

Explanation:

Part A

Step1: Rewrite inequalities to slope-intercept

For $x+2y\geq16$:
$2y\geq -x+16$
$y\geq \frac{-1}{2}x + 8$

For $x+y<20$:
$y < -x + 20$

Step2: Graph boundary lines

• For $y\geq \frac{-1}{2}x + 8$: Use a solid line (since inequality is $\geq$, includes boundary). Plot using intercepts: when $x=0$, $y=8$; when $y=0$, $x=16$.
• For $y < -x + 20$: Use a dashed line (since inequality is $<$, excludes boundary). Plot using intercepts: when $x=0$, $y=20$; when $y=0$, $x=20$.

Step3: Determine shading direction

• For $y\geq \frac{-1}{2}x + 8$: Shade above the solid line (test point $(0,0)$: $0\geq 8$ is false, so shade opposite side).
• For $y < -x + 20$: Shade below the dashed line (test point $(0,0)$: $0 < 20$ is true, so shade this side).
• The solution is the overlapping shaded region.

Part B

Step1: Substitute $(-10,20)$ into inequalities

First inequality: $x+2y\geq16$

$$\begin{align*} -10 + 2(20) &= -10 + 40 = 30 \end{align*}$$

$30\geq16$ is true.

Second inequality: $x+y<20$

$$\begin{align*} -10 + 20 &= 10 \end{align*}$$

$10<20$ is true.

Part C

Step1: Substitute $(20,10)$ into inequalities

First inequality: $x+2y\geq16$

$$\begin{align*} 20 + 2(10) &= 20 + 20 = 40 \end{align*}$$

$40\geq16$ is true.

Second inequality: $x+y<20$

$$\begin{align*} 20 + 10 &= 30 \end{align*}$$

$30<20$ is false.

Answer:

Part A

1. Rewrite both inequalities to slope-intercept form: $y\geq \frac{-1}{2}x + 8$ and $y < -x + 20$.
2. Graph a solid line for $y\geq \frac{-1}{2}x + 8$ (includes boundary) and a dashed line for $y < -x + 20$ (excludes boundary).
3. Shade above the solid line and below the dashed line; the overlapping shaded area is the solution.

Part B

The ordered pair $(-10, 20)$ is in the solution set. It satisfies both inequalities: $-10+2(20)=30\geq16$ and $-10+20=10<20$.

Part C

The ordered pair $(20, 10)$ is NOT in the solution set. While it satisfies $20+2(10)=40\geq16$, it fails $20+10=30<20$, which is false.