Which system of inequalities is represented by the graph?
Understand the Problem
The question asks us to identify the system of inequalities represented by the shaded regions in the given graph. This involves interpreting the lines, their slopes, y-intercepts, and the shaded areas to determine the correct inequalities.
Answer
A. $y > -\frac{1}{2}x + 2$ $y \leq 3x - 1$
Answer for screen readers
A. $y > -\frac{1}{2}x + 2$ $y \leq 3x - 1$
Steps to Solve
- Identify the equations of the lines
From the graph, we can identify two lines. The first line has a negative slope and intersects the y-axis at $y = 2$. The second line has a positive slope and intersects the y-axis at $y = -1$.
- Determine the slope of the first line
The first line passes through the points $(0, 2)$ and $(4, 0)$. The slope $m_1$ is calculated as: $m_1 = \frac{0 - 2}{4 - 0} = \frac{-2}{4} = -\frac{1}{2}$. So, the equation of the first line is $y = -\frac{1}{2}x + 2$.
- Determine the slope of the second line
The second line passes through the points $(0, -1)$ and $(1, 2)$. The slope $m_2$ is calculated as: $m_2 = \frac{2 - (-1)}{1 - 0} = \frac{3}{1} = 3$. So, the equation of the second line is $y = 3x - 1$.
- Determine the inequality for the first line
The first line is dashed, which means the inequality does not include the line itself (i.e., it's either $>$ or $<$). The region above the dashed line is shaded. Therefore, the inequality is $y > -\frac{1}{2}x + 2$.
- Determine the inequality for the second line
The second line is solid, which means the inequality includes the line itself (i.e., it's either $\geq$ or $\leq$). The region below the solid line is shaded. Therefore, the inequality is $y \leq 3x - 1$.
- Combine the Inequalities The system of inequalities is: $y > -\frac{1}{2}x + 2$ $y \leq 3x - 1$
A. $y > -\frac{1}{2}x + 2$ $y \leq 3x - 1$
More Information
The solution is obtained by visually analyzing the graph to deduce the equations of the lines and the regions that satisfy the corresponding inequalities.
Tips
A common mistake is to confuse the direction of the inequality based on which side of the line is shaded. Also, forgetting to consider whether the line is solid or dashed when determining if the inequality includes $\geq$ or $\leq$ versus $>$ or $<$.
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