\begin{tabular}{l} 16. \( y<2 x+1 \) \\ \( y \leq-x-4 \) \\ \hline \\ \hline\end{tabular}

To analyze the system of inequalities given: 1. \( y < 2x + 1 \) 2. \( y \leq -x - 4 \) We can graph these inequalities to find the solution region. ### Step 1: Graph the first inequality \( y < 2x + 1 \) - The line \( y = 2x + 1 \) is a straight line with a slope of 2 and a y-intercept of 1. - Since the inequality is strict (\(<\)), we will use a dashed line to indicate that points on the line are not included in the solution. - The region below this line represents the solution to the inequality. ### Step 2: Graph the second inequality \( y \leq -x - 4 \) - The line \( y = -x - 4 \) has a slope of -1 and a y-intercept of -4. - Since the inequality is non-strict (\(\leq\)), we will use a solid line to indicate that points on the line are included in the solution. - The region below this line represents the solution to the inequality. ### Step 3: Find the intersection of the two inequalities To find the intersection of the two lines, we can set them equal to each other: \[ 2x + 1 = -x - 4 \] Solving for \( x \): \[ 2x + x = -4 - 1 \] \[ 3x = -5 \] \[ x = -\frac{5}{3} \] Now, substitute \( x = -\frac{5}{3} \) back into one of the equations to find \( y \): Using \( y = 2x + 1 \): \[ y = 2\left(-\frac{5}{3}\right) + 1 = -\frac{10}{3} + 1 = -\frac{10}{3} + \frac{3}{3} = -\frac{7}{3} \] So the intersection point is: \[ \left(-\frac{5}{3}, -\frac{7}{3}\right) \] ### Step 4: Determine the solution region The solution region is where the shaded areas of both inequalities overlap. - The area below the dashed line \( y = 2x + 1 \) and below the solid line \( y = -x - 4 \) is the solution to the system of inequalities. ### Conclusion The solution to the system of inequalities \( y < 2x + 1 \) and \( y \leq -x - 4 \) is the region in the coordinate plane that is below both lines, including the line \( y = -x - 4 \) but not including the line \( y = 2x + 1 \). The intersection point \( \left(-\frac{5}{3}, -\frac{7}{3}\right) \) is also part of the boundary of the solution region.