Asked by r
Which of the following graphs represents the solution set of the system of inequalities?y<x3 y≥x2−2
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GPT-4o mini
To analyze the system of inequalities \( y < x^3 \) and \( y \geq x^2 - 2 \), we need to graph both inequalities and find the region where they overlap.
1. **Graphing \( y < x^3 \)**:
- The equation \( y = x^3 \) is a cubic curve that passes through the origin and increases steeply for positive values of \( x \) and decreases for negative values.
- Since it's a strict inequality \( (<) \), we will use a dashed line for the graph of \( y = x^3 \) to indicate that points on the line are not included in the solution set. The region below this curve will be shaded.
2. **Graphing \( y \geq x^2 - 2 \)**:
- The equation \( y = x^2 - 2 \) is a parabola opening upwards, with its vertex located at the point \( (0, -2) \).
- Since it's a non-strict inequality \( (\geq) \), we will use a solid line for the graph of \( y = x^2 - 2 \) to indicate that points on the line are included in the solution set. The region above (and including) this curve will be shaded.
3. **Finding the overlap**:
- The solution set comprises the area where the shaded region for \( y \geq x^2 - 2 \) lies above the curve of \( y = x^2 - 2 \) and below the curve of \( y = x^3 \).
So, to summarize, the solution set represented by the graph should show the area below the dashed line of \( y = x^3 \) and above (or on) the solid line of \( y = x^2 - 2 \).
If you have options with graphs, look for the one that illustrates this intersection correctly!
1. **Graphing \( y < x^3 \)**:
- The equation \( y = x^3 \) is a cubic curve that passes through the origin and increases steeply for positive values of \( x \) and decreases for negative values.
- Since it's a strict inequality \( (<) \), we will use a dashed line for the graph of \( y = x^3 \) to indicate that points on the line are not included in the solution set. The region below this curve will be shaded.
2. **Graphing \( y \geq x^2 - 2 \)**:
- The equation \( y = x^2 - 2 \) is a parabola opening upwards, with its vertex located at the point \( (0, -2) \).
- Since it's a non-strict inequality \( (\geq) \), we will use a solid line for the graph of \( y = x^2 - 2 \) to indicate that points on the line are included in the solution set. The region above (and including) this curve will be shaded.
3. **Finding the overlap**:
- The solution set comprises the area where the shaded region for \( y \geq x^2 - 2 \) lies above the curve of \( y = x^2 - 2 \) and below the curve of \( y = x^3 \).
So, to summarize, the solution set represented by the graph should show the area below the dashed line of \( y = x^3 \) and above (or on) the solid line of \( y = x^2 - 2 \).
If you have options with graphs, look for the one that illustrates this intersection correctly!
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