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Magazine Chapter 6: Problem 31
Use a graph to solve the given inequality. $$ e^{x-2}<1 $$
Short Answer
Expert verified
The solution is \(x < 2\).
Step by step solution
01
Understand the inequality
The inequality given is \(e^{x-2} < 1\). The expression \(e^{x-2}\) involves a transformation of the natural exponential function, and it needs to be less than 1.
02
Simplify the inequality
We know that for \(e^{x-2}\) to be less than 1, \(x-2\) must be less than 0. This is because the exponential function \(e^u\) is equal to 1 when \(u = 0\), greater than 1 when \(u > 0\), and less than 1 when \(u < 0\). Thus, we solve \(x-2 < 0\).
03
Solve the resulting inequality
Add 2 to both sides of \(x-2 < 0\) to isolate \(x\):\[x < 2.\]So, the solution to the inequality is \(x < 2\).
04
Graph the inequality
To graph \(x < 2\), draw a number line. Plot a point at \(x = 2\) with an open circle to indicate that 2 is not included in the solution set. Shade the line to the left of 2, indicating all values less than 2 are solutions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphical Solutions
When solving inequalities, graphical solutions can provide a clear visual representation of the solution set. Graphing helps us quickly identify the range of values that satisfy the inequality.
To solve the inequality graphically,
To solve the inequality graphically,
- First, consider the function involved, which in this case is the exponential function.
- Plot the function. For the given inequality, plot the graph of \(e^{x-2}\) and the line \(y = 1\).
- The solution to the inequality \(e^{x-2} < 1\) is where the graph of the function is below the line \(y = 1\).
- On the graph, identify where the function, \(e^{x-2}\), crosses or is below the horizontal line \(y = 1\). In this specific example, this will correspond to the section of the x-axis where the graph is below the line, which is when \(x < 2\).
Exponential Functions
An exponential function involves a constant base raised to a variable exponent. In mathematical terms, it typically looks like \(e^{x}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
Exponential functions grow rapidly. A small change in the exponent produces a large change in the function's value.
Key aspects of exponential functions include:
Exponential functions grow rapidly. A small change in the exponent produces a large change in the function's value.
Key aspects of exponential functions include:
- They always have a graph that passes through the point (0,1), since \(e^0 = 1\).
- These functions approach zero as \(x\) approaches negative infinity but never actually reach zero.
- When transformed, such as \(e^{x-2}\), it shifts the basic curve horizontally by changing the exponent \(x\) to \(x-2\). This translates the graph 2 units to the right.
Inequality Solution Steps
Solving inequalities can be broken down into a series of manageable steps. In our example, solving the inequality \(e^{x-2} < 1\) requires a few key steps:
1. **Understanding the inequality**:
1. **Understanding the inequality**:
- Recognize the form of the inequality and the function involved.
- In our case, identify that \(e^{x-2}\) should be less than 1.
- Knowing the behavior of exponential functions, propose that \(x-2 < 0\) because exponential terms are less than 1 when their exponent is negative.
- Manipulate the inequality \(x-2 < 0\) to isolate \(x\).
- Add 2 to both sides, resulting in \(x < 2\).
- Draw the solution on a number line or using the graph of the function for a visual verification.
- An open circle on 2 means it's not included, and shading to the left shows all \(x\) values that solve the inequality.
