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 3 of 3 : graph the solution set for the system.\
answer

Explanation:

Step1: Graph boundary $y=-2x+3$

Draw a solid line (since $\leq$ includes equality) with slope $-2$ and y-intercept $(0,3)$. Shade the region below this line (because $y \leq -2x+3$ means all points where $y$ is less than or equal to the line's values).

Step2: Graph boundary $y=6x-9$

Draw a dashed line (since $>$ does not include equality) with slope $6$ and y-intercept $(0,-9)$. Shade the region above this line (because $y > 6x-9$ means all points where $y$ is greater than the line's values).

Step3: Find intersection of lines

Set $-2x+3 = 6x-9$:

$$\begin{align*} -2x -6x &= -9 -3 \\ -8x &= -12 \\ x &= \frac{-12}{-8} = \frac{3}{2} = 1.5 \end{align*}$$

Substitute $x=1.5$ into $y=-2x+3$:
$y = -2(1.5)+3 = -3+3=0$
Intersection point: $(1.5, 0)$

Step4: Identify solution set

The solution is the overlapping shaded region (below the solid line $y=-2x+3$ and above the dashed line $y=6x-9$), bounded by the two lines and meeting at $(1.5, 0)$.

Answer:

The solution set is the overlapping shaded region: below the solid line $y=-2x+3$, above the dashed line $y=6x-9$, with the boundary lines intersecting at $(1.5, 0)$.