Question

solve the system of two linear inequalities graphically.\

$$\begin{cases} y \\leq -2x + 3 \\\\ y > 6x - 9 \\end{cases}$$

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step 2 of 3 : graph the solution set of the second linear inequality.

Explanation:

Step1: Identify boundary line

The boundary line is $y=6x-9$. Since the inequality is $y>6x-9$, the line is dashed (not included in the solution).

Step2: Find intercepts for plotting

• x-intercept: Set $y=0$, solve $0=6x-9$

$6x=9 \implies x=\frac{3}{2}=1.5$
So intercept is $(1.5, 0)$

• y-intercept: Set $x=0$, solve $y=6(0)-9$

$y=-9$
So intercept is $(0, -9)$

Step3: Test a point to shade region

Use test point $(0,0)$: $0 > 6(0)-9 \implies 0 > -9$, which is true. Shade the region containing $(0,0)$ (above the dashed line).

Answer:

1. Draw a dashed line through the points $(1.5, 0)$ and $(0, -9)$ (this is $y=6x-9$).
2. Shade the region above this dashed line (the area where all points satisfy $y > 6x - 9$).