What inequality does the graph represent?

Ask by King Park. in the United States

Jan 24,2025

Question

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

UpStudy AI · Accepted Answer

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 $$ y = x + 2 $$. The graph represents the inequalities $$ y > \frac{1}{4}x + 2 $$ and $$ y < x + 2 $$.

What inequality does the graph represent?

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