Question

use the drawing tool(s) to form the correct answer on the provided graph. graph the solution to the following system of inequalities in the coordinate plane. 2x - 3y < 15 y ≤ x + 2

Explanation:

Step1: Rewrite first inequality to slope-intercept form

Rearrange $2x - 3y < 15$:
$-3y < -2x + 15$
$y > \frac{2}{3}x - 5$
(Note: Inequality flips when dividing by negative)

Step2: Identify line type for first inequality

Since $y > \frac{2}{3}x - 5$ uses $>$, draw a dashed line for $y = \frac{2}{3}x - 5$.
Find intercepts:

• x-intercept: Set $y=0$: $0 = \frac{2}{3}x - 5 \implies x = 7.5$
• y-intercept: Set $x=0$: $y = -5$

Step3: Rewrite second inequality to slope-intercept form

$y \leq x + 2$ is already in slope-intercept form.

Step4: Identify line type for second inequality

Since $y \leq x + 2$ uses $\leq$, draw a solid line for $y = x + 2$.
Find intercepts:

• x-intercept: Set $y=0$: $0 = x + 2 \implies x = -2$
• y-intercept: Set $x=0$: $y = 2$

Step5: Determine shaded region

1. For $y > \frac{2}{3}x - 5$: Shade above the dashed line.
2. For $y \leq x + 2$: Shade below the solid line.

The solution is the overlapping shaded region.

Answer:

1. Draw a dashed line through points $(7.5, 0)$ and $(0, -5)$ (for $y = \frac{2}{3}x - 5$).
2. Draw a solid line through points $(-2, 0)$ and $(0, 2)$ (for $y = x + 2$).
3. Shade the region that is above the dashed line and below the solid line.