Linear Inequalities Graph C7E9Ab

1. **State the problem:** We need to graph the system of linear inequalities: $$\begin{cases} y \leq -5x + 2 \\ y \leq -\frac{3}{2}x - 1 \end{cases}$$ and identify the correct shaded region that represents the solution. 2. **Understand the inequalities:** Each inequality represents a half-plane bounded by the line: - Line 1: $$y = -5x + 2$$ - Line 2: $$y = -\frac{3}{2}x - 1$$ The inequalities are \(y \leq\) these lines, so the solution is the region below or on each line. 3. **Analyze the boundary lines:** - The first line has slope $$-5$$ and y-intercept $$2$$. - The second line has slope $$-\frac{3}{2}$$ and y-intercept $$-1$$. 4. **Find the intersection point of the two lines:** Set $$-5x + 2 = -\frac{3}{2}x - 1$$ Multiply both sides by 2 to clear fraction: $$2(-5x + 2) = 2\left(-\frac{3}{2}x - 1\right)$$ $$-10x + 4 = -3x - 2$$ Bring all terms to one side: $$-10x + 4 + 3x + 2 = 0$$ $$-7x + 6 = 0$$ $$-7x = -6$$ $$x = \frac{6}{7}$$ Substitute back to find $$y$$: $$y = -5\left(\frac{6}{7}\right) + 2 = -\frac{30}{7} + 2 = -\frac{30}{7} + \frac{14}{7} = -\frac{16}{7}$$ So intersection point is $$\left(\frac{6}{7}, -\frac{16}{7}\right)$$. 5. **Determine the solution region:** Since both inequalities are $$y \leq$$ their respective lines, the solution is the intersection of the half-planes below both lines. 6. **Check the shaded regions described:** - Left graph: shaded region is above-left of both lines — this is above the lines, so it does not satisfy $$y \leq$$. - Middle graph: shaded region is to the right of the blue line and above the red line — again above lines, so no. - Right graph: shaded region is below both lines, mainly on the left side — this matches $$y \leq$$ both lines. 7. **Conclusion:** The correct graph is the right graph where the shaded region is below both lines. --- **Final answer:** The solution region is the intersection of the half-planes below the lines $$y = -5x + 2$$ and $$y = -\frac{3}{2}x - 1$$, which corresponds to the right graph described.