Which graph represents the system of inequalities?
Understand the Problem
The question is asking which graph accurately represents the system of inequalities given by two equations. The inequalities need to be analyzed to determine their graphical representation.
Answer
The shaded area representing both inequalities above the line \( y = -\frac{1}{2}x + 2 \) and below the line \( y = \frac{1}{3}x - 1 \).
Answer for screen readers
The graph that accurately represents the system of inequalities shows the shaded area above the line ( y = -\frac{1}{2}x + 2 ) and below the line ( y = \frac{1}{3}x - 1 ).
Steps to Solve
- Identify the equations of the lines
Convert the inequalities into equations to find the boundary lines:
- For the first inequality, set ( y = -\frac{1}{2}x + 2 ).
- For the second inequality, set ( y = \frac{1}{3}x - 1 ).
- Graph the first line
Plot the line for the first equation ( y = -\frac{1}{2}x + 2 ) using the slope and y-intercept.
- The y-intercept is 2, and the slope is (-\frac{1}{2}), meaning for every 1 unit moved right, go down 0.5 units.
- Shade the correct region for the first inequality
Since the inequality is ( y > -\frac{1}{2}x + 2 ), shade the area above the line, as this indicates the values of ( y ) that are greater.
- Graph the second line
Plot the line for the second equation ( y = \frac{1}{3}x - 1 ).
- The y-intercept is -1, and the slope is (\frac{1}{3}), meaning for every 1 unit moved right, go up 1/3 unit.
- Shade the correct region for the second inequality
Since the inequality is ( y < \frac{1}{3}x - 1 ), shade the area below this line, indicating the values of ( y ) that are less.
- Determine the intersection of the shaded regions
The solution to the system of inequalities will be where the shading from both inequalities overlaps. This will give the area that satisfies both conditions.
The graph that accurately represents the system of inequalities shows the shaded area above the line ( y = -\frac{1}{2}x + 2 ) and below the line ( y = \frac{1}{3}x - 1 ).
More Information
The graph represents the solution set where both inequalities are true. The area of overlap is critical in determining the valid solutions to the system.
Tips
- Failing to identify the correct direction of shading for each inequality (above vs. below the line).
- Confusing the slopes or intercepts which can lead to incorrect graph placements.
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