What inequality does the graph represent? \( \begin{array}{l}yx+2 \\ y<\frac{1}{4} x+2 \\ y>\frac{1}{4} z+2\end{array} \)

To determine which inequality the graph represents, let's analyze the given set of inequalities: \[ \begin{array}{l} 1. \quad y < x + 2 \\ 2. \quad y > x + 2 \\ 3. \quad y < \frac{1}{4}x + 2 \\ 4. \quad y > \frac{1}{4}x + 2 \\ \end{array} \] Assuming there's a typo in the fourth inequality (changing \( z \) to \( x \)), we can consider the inequalities as: \[ \begin{array}{l} 1. \quad y < x + 2 \quad \text{(Region below the line \( y = x + 2 \))} \\ 2. \quad y > x + 2 \quad \text{(Region above the line \( y = x + 2 \))} \\ 3. \quad y < \frac{1}{4}x + 2 \quad \text{(Region below the line \( y = \frac{1}{4}x + 2 \))} \\ 4. \quad y > \frac{1}{4}x + 2 \quad \text{(Region above the line \( y = \frac{1}{4}x + 2 \))} \\ \end{array} \] ### Understanding the Graph 1. **Boundary Lines:** - **\( y = x + 2 \):** This is a straight line with a slope of 1 and a y-intercept at (0, 2). - **\( y = \frac{1}{4}x + 2 \):** This line has a gentler slope of \( \frac{1}{4} \) and the same y-intercept at (0, 2). 2. **Shading Regions:** - **\( y < x + 2 \):** Shaded area below the line \( y = x + 2 \). - **\( y > x + 2 \):** Shaded area above the line \( y = x + 2 \). - **\( y < \frac{1}{4}x + 2 \):** Shaded area below the line \( y = \frac{1}{4}x + 2 \). - **\( y > \frac{1}{4}x + 2 \):** Shaded area above the line \( y = \frac{1}{4}x + 2 \). ### Possible Scenarios - **Single Inequality Representation:** - If the graph shows a single shaded region relative to one boundary line, it represents one of the four inequalities listed above. - **System of Inequalities:** - If the graph shows an intersection of two shaded regions (creating a bounded area), it likely represents a system of two inequalities. - For example, if the shaded area is between \( y = \frac{1}{4}x + 2 \) and \( y = x + 2 \), the system would be: \[ \frac{1}{4}x + 2 < y < x + 2 \] Which corresponds to: \[ y > \frac{1}{4}x + 2 \quad \text{and} \quad y < x + 2 \] ### Conclusion Without seeing the specific graph, the most comprehensive interpretation is that it represents a **system of inequalities** where \( y \) is bounded between \( \frac{1}{4}x + 2 \) and \( x + 2 \). Therefore, the graph likely represents: \[ \frac{1}{4}x + 2 < y < x + 2 \] **Or equivalently:** \[ y > \frac{1}{4}x + 2 \quad \text{and} \quad y < x + 2 \] This means the graph shades the region **above** the line \( y = \frac{1}{4}x + 2 \) **and below** the line \( y = x + 2 \).