Question

1. the shaded portion of which graph best represents the solution to the inequality (2y - x geq 4)? four graph options with selection circles are present below the question

Explanation:

Answer:

To solve \( 2y - x \geq 4 \), first rewrite it in slope - intercept form (\( y=mx + b\)):

1. Add \( x \) to both sides: \( 2y\geq x + 4 \)
2. Divide both sides by 2: \( y\geq\frac{1}{2}x + 2 \)

Step 1: Analyze the boundary line

The equation of the boundary line is \( y=\frac{1}{2}x + 2 \). The slope \( m=\frac{1}{2}\) and the \( y\) - intercept \( b = 2\). Since the inequality is \( y\geq\frac{1}{2}x + 2 \), the boundary line should be solid (because the inequality includes equality, \( \geq\)).

Step 2: Analyze the shading

We test a point not on the line, say the origin \((0,0)\):

Substitute \( x = 0\) and \( y = 0\) into the inequality \( y\geq\frac{1}{2}x + 2\):

\( 0\geq\frac{1}{2}(0)+2\), which simplifies to \( 0\geq2\). This is false. So we shade the region that does not contain the origin.

Now, let's analyze the options:

• For a line \( y=\frac{1}{2}x + 2\), when \( x = 0\), \( y = 2\) (the \( y\) - intercept), and when \( y = 0\), \( 0=\frac{1}{2}x+2\Rightarrow\frac{1}{2}x=-2\Rightarrow x=-4\) (the \( x\) - intercept is \((-4,0)\)).
• The boundary line is solid (because of \( \geq\)). And we shade above the line (since the test point \((0,0)\) gives a false statement, we shade the opposite side of the origin relative to the line).

Looking at the given graphs, the graph where the boundary line \( y = \frac{1}{2}x+2\) (with \( y\) - intercept 2 and \( x\) - intercept - 4) is solid and the region above the line is shaded (and does not include the origin) is the correct one. If we assume the first graph (the top - most one) has a solid line \( y=\frac{1}{2}x + 2\) and shades the region above the line (including the area that does not have the origin), then that is the correct graph. But since we can't see the exact labels, based on the inequality \( y\geq\frac{1}{2}x + 2\), the graph with a solid line \( y=\frac{1}{2}x + 2\) and shading above the line (not including the origin) is the answer. If we have to choose from the given options (assuming the first option is the one with the correct solid line and shading above), the answer is the first graph (the top - most circular - optioned graph). But since the user's image is a bit unclear, but following the steps, the graph with solid line \( y=\frac{1}{2}x + 2\) and shading above the line (not containing \((0,0)\)) is the solution.

(Note: If we had to pick from the options, and assuming the first option is the one with the correct solid line and shading, the answer would be the first graph. But due to the image quality, we can be more precise with the steps above. However, if we assume the options are labeled as A, B, C, D (top to bottom), and the first option (A) has a solid line \( y=\frac{1}{2}x + 2\) and shades above the line, then the answer is A. The graph with a solid boundary line \( y=\frac{1}{2}x + 2\) (passing through \((0,2)\) and \((-4,0)\)) and shading the region above the line (not including the origin) is the correct representation of \( 2y - x\geq4\).)

If we have to write the final answer as the graph (assuming the first graph is the correct one), we can say that the graph with the solid line \( y=\frac{1}{2}x + 2\) (with \( y\) - intercept 2 and \( x\) - intercept - 4) and shading above the line (not containing the origin) is the solution. But since the user's problem is about choosing the graph, and if we assume the first option is correct, the answer is the first graph (the top - most one among the four options).

(If we use the standard way, after rewriting the inequality as \( y\geq\frac{1}{2}x + 2\), the graph should have a solid line (because of \(\geq\)) with slope \( \frac{1}{2}\) and \( y\) - intercept 2, and shading above the line. So the graph that matches this description is the answer. If we consider the options, and assuming the first option is the one with this property, the answer is the first graph.)