Question

nc math 1—released items
2 in which graph does the shaded region represent the solution set for the inequality shown below?
2x + y < 4
four graphs labeled a, b, c, d with shaded regions
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Explanation:

Step1: Rewrite the inequality

First, rewrite the inequality \(2x + y < 4\) in slope - intercept form (\(y=mx + b\)) by solving for \(y\).
Subtract \(2x\) from both sides: \(y<-2x + 4\).

Step2: Analyze the boundary line

The boundary line for the inequality \(y=-2x + 4\) is a straight line. Since the inequality is \(y < - 2x+4\) (not \(y\leq - 2x + 4\)), the boundary line should be a dashed line (to indicate that the points on the line are not included in the solution set).

• The \(y\) - intercept of the line \(y=-2x + 4\) is \(b = 4\), so the line crosses the \(y\) - axis at \((0,4)\).
• The slope \(m=-2\), which means for every 1 unit we move to the right along the \(x\) - axis, we move down 2 units along the \(y\) - axis. So, when \(x = 2\), \(y=-2(2)+4=0\), so the line also passes through \((2,0)\).

Step3: Determine the shaded region

We need to determine which side of the line \(y=-2x + 4\) to shade. We can use a test point, for example, the origin \((0,0)\).
Substitute \(x = 0\) and \(y = 0\) into the inequality \(y<-2x + 4\):
\(0<-2(0)+4\), which simplifies to \(0 < 4\). This is a true statement. So, we shade the region that contains the origin \((0,0)\).

Now let's analyze each option:

• Option A: The boundary line seems to be solid (incorrect, since the inequality is strict), and the shaded region does not seem to contain the origin (incorrect).
• Option B: The boundary line is dashed? Wait, no, looking at the graph, the line in B is solid? Wait, no, the inequality is \(y < - 2x + 4\), the boundary should be dashed. But also, when we test the origin \((0,0)\) in the region of B, the region in B is above the line \(y=-2x + 4\) (since for \(x = 0,y\) in the shaded region of B is greater than 4). Substituting \((0,0)\) into \(y < - 2x+4\) gives \(0 < 4\), but the shaded region in B is above the line, so it does not satisfy the inequality (incorrect).
• Option C: The boundary line is in the wrong place (the slope and intercept do not match \(y=-2x + 4\)) (incorrect).
• Option A: Wait, let's re - check. Wait, maybe I made a mistake. Wait, the line \(y=-2x + 4\) has a \(y\) - intercept of 4 and \(x\) - intercept of 2. The shaded region in option A: Let's check the test point \((0,0)\). In option A, the shaded region includes \((0,0)\), and the boundary line is dashed? Wait, the graph in A: the line goes from \((0,4)\) to \((2,0)\) (dashed or solid? From the image, it seems like the line in A is dashed? Wait, no, the original problem's option A: the shaded region is below the line \(y=-2x + 4\) (since when \(x = 0,y\) in the shaded region is less than 4, and when \(x = 2,y\) is less than 0). Wait, no, let's re - express the inequality \(2x + y<4\) or \(y < - 2x + 4\). The correct graph should have a dashed line through \((0,4)\) and \((2,0)\) and shade the region below the line (since \(y\) is less than \(-2x + 4\)).

Looking at the options:

• Option A: The line connects \((0,4)\) and \((2,0)\), and the shaded region is below the line (contains \((0,0)\)) and the line is dashed (since the inequality is strict).
• Option B: The shaded region is above the line (incorrect, because \(y < - 2x+4\) means below the line).
• Option C: The line is not \(y=-2x + 4\) (wrong slope and intercept).
• Option D: The line is not \(y=-2x + 4\) (wrong slope and intercept).

Answer:

A