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y - 1 < (1/2)(x + 3) y - 2 > (1/2)(x + 1) y + 1 ≤ (1/2)(x - 3) y + 2 ≥ …

Question

y - 1 < (1/2)(x + 3)
y - 2 > (1/2)(x + 1)
y + 1 ≤ (1/2)(x - 3)
y + 2 ≥ (1/2)(x - 1)

Explanation:

To solve this, we analyze the graph's line (slope, intercept) and the inequality's shading (dashed/solid line, direction).

Step 1: Find the line’s equation

The line has a slope \( m = \frac{1}{2} \) (rise over run: \( \frac{1}{2} \)). From the graph, a point on the line is \( (-3, 1) \) (or similar). Using point - slope form \( y - y_1 = m(x - x_1) \):
For point \( (-3, 1) \), \( y - 1=\frac{1}{2}(x + 3) \).

Step 2: Analyze the inequality type

The line is dashed (so inequality is strict: \( < \) or \( > \)), and the shading is above/below? Wait, re - check. Wait, the first option is \( y - 1<\frac{1}{2}(x + 3) \)? Wait no, wait the graph’s line: let's re - evaluate. Wait, the line passes through \( (-3,1) \) and has slope \( \frac{1}{2} \). Let's check the options:

  • Option 1: \( y - 1<\frac{1}{2}(x + 3) \). The line is \( y - 1=\frac{1}{2}(x + 3) \). The shading: if we test a point in the shaded region (e.g., \( (0,3) \)):

Left - hand side: \( 3 - 1 = 2 \). Right - hand side: \( \frac{1}{2}(0 + 3)=\frac{3}{2}=1.5 \). \( 2<1.5 \)? No. Wait, maybe I mixed up. Wait, maybe the line is \( y + 2=\frac{1}{2}(x - 1) \)? No, let's re - check the options. Wait, the correct approach:

  1. Slope and Intercept Check:

The slope of all options is \( \frac{1}{2} \). Now, find the y - intercept or a point. Let's rewrite each option in slope - intercept (\( y=mx + b \)):

  • Option 1: \( y - 1<\frac{1}{2}(x + 3)\Rightarrow y<\frac{1}{2}x+\frac{3}{2}+1\Rightarrow y<\frac{1}{2}x+\frac{5}{2} \).
  • Option 2: \( y - 2>\frac{1}{2}(x + 1)\Rightarrow y>\frac{1}{2}x+\frac{1}{2}+2\Rightarrow y>\frac{1}{2}x+\frac{5}{2} \).
  • Option 3: \( y + 1\leq\frac{1}{2}(x - 3)\Rightarrow y\leq\frac{1}{2}x-\frac{3}{2}-1\Rightarrow y\leq\frac{1}{2}x-\frac{5}{2} \).
  • Option 4: \( y + 2\geq\frac{1}{2}(x - 1)\Rightarrow y\geq\frac{1}{2}x-\frac{1}{2}-2\Rightarrow y\geq\frac{1}{2}x-\frac{5}{2} \).
  1. Test a point in the shaded region:

From the graph, the shaded region includes points like \( (0,3) \), \( ( - 5,0) \), etc. Let's test \( (0,3) \) in each option:

  • Option 1: \( 3 - 1<\frac{1}{2}(0 + 3)\Rightarrow2<1.5 \)? False.
  • Option 2: \( 3 - 2>\frac{1}{2}(0 + 1)\Rightarrow1>\frac{1}{2} \)? True. Wait, but earlier slope - point was wrong. Wait, maybe the line passes through \( (-1,2) \). Let's recalculate slope: from \( (-1,2) \) to \( (1,3) \), slope is \( \frac{3 - 2}{1 - (-1)}=\frac{1}{2} \). Then point - slope for \( (-1,2) \): \( y - 2=\frac{1}{2}(x + 1) \). Now, the inequality: the line is dashed (so strict), and shading is above? Wait, option 2 is \( y - 2>\frac{1}{2}(x + 1) \). Testing \( (0,3) \): \( 3 - 2 = 1 \), \( \frac{1}{2}(0 + 1)=0.5 \), \( 1>0.5 \), which is true. Wait, but earlier mistake in point selection. Wait, the original graph: let's check the options again. Wait, the first option’s line: \( y - 1=\frac{1}{2}(x + 3) \) passes through \( (-3,1) \) and \( ( - 1,2) \) (since \( y - 1=\frac{1}{2}(x + 3) \), when \( x=-1 \), \( y - 1=\frac{1}{2}(2)=1\Rightarrow y = 2 \)), which matches the line in the graph. Now, the shading: the region is above or below? Wait, the inequality \( y - 1<\frac{1}{2}(x + 3) \) would shade below the line, but the graph’s shading is above? Wait, no, maybe I messed up. Wait, let's take a point in the shaded area, say \( (0,3) \). Plug into \( y - 1<\frac{1}{2}(x + 3) \): \( 3 - 1 = 2 \), \( \frac{1}{2}(0 + 3)=1.5 \), \( 2<1.5 \) is false. Plug into \( y - 2>\frac{1}{2}(x + 1) \): \( 3 - 2 = 1 \), \( \frac{1}{2}(0 + 1)=0.5 \), \( 1>0.5 \) is true. Wait, but the line for option 2 is \( y -…

Answer:

B. \( y - 2>\frac{1}{2}(x + 1) \) (assuming the options are labeled A, B, C, D with B as \( y - 2>\frac{1}{2}(x + 1) \))