Question

algebra 2a semester review back to intro page click for long desc when the system of inequalities is solved by graphing, which point represents a solution to the system? (1 point) (22, -8) (-4, -8) (10, 8) (4, -4)

Explanation:

Step1: Analyze the feasible region

The solution to a system of inequalities is the region where all inequalities overlap (the doubly - shaded region, which is region C and D? Wait, no, the doubly - shaded region here is the area that is in both the orange region and the striped region. Looking at the graph, the feasible region (solution region) is the area that is in both the left - hand (orange) region and the striped region. So we need to check which of the given points lies in this overlapping region.

Step2: Check each point

- For point \((22,-8)\): The x - coordinate is 22, which is far to the right. The left - hand region (orange) seems to be for \(x\leq - 4\) (since the vertical line is at \(x=-4\) maybe? Wait, looking at the graph, the orange region is to the left of a vertical line (probably \(x = - 4\) or something? Wait no, the orange region is on the left, and the striped region is below the dashed line. Wait, let's re - evaluate. Wait, the vertical line: the orange region is to the left of a vertical line (let's say \(x=-4\) as the boundary? Wait, no, the orange region is from the left up to a vertical line (maybe \(x = - 4\)?). Wait, the point \((-4,-8)\): x - coordinate is - 4, y - coordinate is - 8. Wait, no, let's check the x - values. The first point: \((22,-8)\): x = 22 is in the right - hand side, outside the orange (left) region. So it's not in the solution.
- For point \((-4,-8)\): Let's see the x - coordinate is - 4 (on the boundary of the orange region) and y - coordinate is - 8. The dashed line: let's find the equation of the dashed line. The dashed line passes through points, let's assume from the graph, when x = 0, y=-4? Wait, no, looking at the graph, the dashed line goes from, say, when x=-4, y = - 8? Wait, no, let's check the point \((-4,-8)\): x=-4 (on the vertical boundary of the orange region) and y = - 8. Wait, the striped region is below the dashed line. Let's check the y - value. The dashed line: let's find its slope. From the graph, when x = 0, y=-4? Wait, no, maybe the dashed line has a slope of \(\frac{1}{2}\) or something. Wait, alternatively, let's check the x - coordinate of each point. The orange region is to the left of a vertical line (let's say \(x=-4\))? Wait, no, the point \((-4,-8)\): x=-4, which is on the vertical boundary of the orange region (if the orange region is \(x\leq - 4\)) and the striped region (below the dashed line). Let's check the other points:
- Point \((10,8)\): x = 10 is in the right - hand side, outside the orange region.
- Point \((4,-4)\): x = 4 is in the right - hand side, outside the orange region.
- Point \((-4,-8)\): x=-4 (in the orange region, since the orange region is to the left of or on the vertical line at \(x = - 4\)) and y=-8. Now, check if it's below the dashed line. The dashed line: let's see, when x = 0, y=-4? If the slope is \(\frac{y_2 - y_1}{x_2 - x_1}\), if we take two points on the dashed line, say (0, - 4) and (5, - 1.5)? No, maybe better to check the y - value. The point \((-4,-8)\): y=-8. The dashed line at x=-4: what's the y - value? If the dashed line passes through (0, - 4) and (5, - 1), the slope \(m=\frac{-1+4}{5 - 0}=\frac{3}{5}\). Then the equation is \(y=\frac{3}{5}x-4\). At x=-4, \(y=\frac{3}{5}(-4)-4=-\frac{12}{5}-4=-\frac{12 + 20}{5}=-\frac{32}{5}=-6.4\). Wait, but - 8 < - 6.4, so the point \((-4,-8)\) is below the dashed line. And x=-4 is in the orange region (left of the vertical line? Wait, maybe the vertical line is x=-4, and the orange region is \(x\leq - 4\)). So \((-4,-8)\) is in the overlapping region (the solution region). Wait, but let's re - check the options. Wait, the options are \((22,-8)\), \((-4,-8)\), \((10,8)\), \((4,-4)\).

Wait, maybe I made a mistake in the vertical line. Let's look at the x - coordinates. The orange region is on the left, so x is less than or equal to a certain value. The point \((-4,-8)\): x=-4, which is in the orange region (since the orange region includes x=-4, maybe the vertical line is x=-4, and the orange region is \(x\leq - 4\)). The striped region is below the dashed line. So for \((-4,-8)\): x=-4 (in orange region), y=-8 (below the dashed line, since the dashed line at x=-4 has a higher y - value than - 8). So it's in the solution region.

For \((10,8)\): x = 10 is in the right - hand side, outside the orange region. So not in solution.

For \((4,-4)\): x = 4 is in the right - hand side, outside the orange region. Not in solution.

For \((22,-8)\): x = 22 is in the right - hand side, outside the orange region. Not in solution.

Answer:

\((-4,-8)\) (the option with \((-4,-8)\))