Full solution

\left\{\begin{array}{l} y \geq-\frac{1}{3} x+3 \\ y>\frac{3}{4} x-1 \end{array}\right.

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Which quadrants contain the solution to this system of inequalities?

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A. quadrants I, II, and IV

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B. quadrants I and II

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C. quadrants II and III

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D. quadrants I and IV

Analyze Line Inequality: Analyze the first inequality $y \geq -\frac{1}{3}x+3$ . This inequality represents a line with a negative slope that crosses the y-axis at $(0, 3)$ . Since $y$ is greater than or equal to this line, the solution set is above the line. This line divides the coordinate plane into two regions: above the line (where the inequality holds) and below the line (where the inequality does not hold).
Analyze Positive Slope: Analyze the second inequality $y > \frac{3}{4}x-1$ . This inequality represents a line with a positive slope that crosses the y-axis at $(0, -1)$ . Since $y$ is greater than this line, the solution set is above the line. This line also divides the coordinate plane into two regions: above the line (where the inequality holds) and below the line (where the inequality does not hold).
Determine Common Quadrants: Determine the quadrants where both inequalities are satisfied. $\newline$ The first inequality's solution set is above the line with a negative slope, which includes parts of quadrants $I$ , $II$ , and $III$ . The second inequality's solution set is above the line with a positive slope, which includes parts of quadrants $I$ and $II$ . The common regions where both inequalities are satisfied are in quadrants $I$ and $II$ .
Confirm Excluded Quadrants: Confirm that no other quadrants are included in the solution set. Quadrant III is not included because the first inequality requires $y$ to be above the line with a negative slope, which does not occur in quadrant III. Quadrant IV is not included because the second inequality requires $y$ to be above the line with a positive slope, which does not occur in quadrant IV.