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Magazine Chapter 17: Problem 30
Draw a sketch of the graph of the region in which the points satisfy the given
system of inequalities.
$$\begin{aligned}
&y>0\\\
&y>1-x\\\
&y
Short Answer
Expert verified
The solution region forms a triangular area above the x-axis, above the line \( y = 1-x \), and below the curve \( y = e^x \).
Step by step solution
01
Understand the Inequalities
We have three inequalities to sketch:1. **\( y > 0 \)**: This represents the region above the x-axis.2. **\( y > 1-x \)**: This indicates the area above the line \( y = 1-x \), which is a line with a y-intercept of 1 and a slope of -1.3. **\( y < e^x \)**: This implies the region below the curve \( y = e^x \), which is an exponential growth function.
02
Sketch the Boundary Lines and Curves
Draw the boundary lines and curve without inequality signs:- Draw the x-axis to represent \( y = 0 \).- Draw the line \( y = 1-x \) by marking the y-intercept at (0, 1) and noticing it has a slope of -1. - Draw the curve \( y = e^x \), noting that it passes through (0,1) and rises rapidly for positive x-values.
03
Determine the Feasible Region
Identify where the inequalities overlap:- Shaded region is above both the x-axis and the line \( y = 1-x \).- Shaded region is below the curve \( y = e^x \).- This overlapping region forms the feasible area where all inequalities are true, starting from below the curve but above both the x-axis and line \( y = 1-x \).
04
Sketch the Combined Region
Carefully shade the overlapping region determined in Step 3.- Ensure the shading is above the x-axis, above the line \( y = 1-x \), and below the curve \( y = e^x \).- The correct area should look like a triangular region bounded at the base by the x-axis and its sides between the line and curve.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
System of Inequalities
A system of inequalities involves multiple inequalities that dictate conditions for solutions. In this exercise, we consider three inequalities: \(y > 0\), \(y > 1-x\), and \(y < e^{x}\). Each of these represents different parts of the coordinate plane.
- The first inequality, \(y > 0\), simply means that \(y\) is positive, so we look for points above the x-axis.
- The second, \(y > 1-x\), describes a half-plane above the line \(y = 1-x\). This line has a y-intercept of 1 and a slope of -1, slanting downwards to the right.
- The third inequality, \(y < e^{x}\), places our interest below the curve of the exponential function \(y = e^{x}\), which grows rapidly as \(x\) increases.
Graph Sketching
Graph sketching is a valuable skill in understanding mathematical inequalities and solutions. For this system, graph sketching involves:
- First plotting the boundary line for each inequality without considering the inequality signs, which are just `>` or `<`. This means drawing the lines and curves as if they are equations (\(y =\)).
- The x-axis represents \(y = 0\). Drawing this helps visualize where \(y\) is above zero.
- The linear boundary \(y = 1-x\) is plotted by taking key points, such as the y-intercept (0,1) and another point by intercepting the x-axis at \(x = 1\).
- The exponential curve \(y = e^{x}\) starts at \(x = 0\) with a \(y\) of 1 and rises steeply for positive \(x\) values.
Feasible Region
The feasible region is the area in the graph that satisfies all the inequalities of a system simultaneously. Identifying this region entails:
- Locating where all the different shaded areas from each inequality union intersect.
- Checking points within each possible region against all inequalities to ensure correctness.
- In this exercise, the feasible region lies above the x-axis, above this linear line \(y = 1-x\), and simultaneously below the exponential curve \(y = e^{x}\).
Exponential and Linear Functions
Understanding the interaction between exponential and linear functions is critical in graphing and systems of inequalities.
- Linear functions, like \(y = 1-x\), produce straight lines. These lines have constant slopes and define half-planes when used in inequalities.
- Exponential functions, such as \(y = e^{x}\), exhibit growth patterns that start slowly and increase exponentially, giving a curved appearance.
- In graphing inequalities, it's crucial to recognize where a line crosses or stays within an exponential curve. This intersection often defines the boundaries of the feasible region.
