Problem 31 - Precalculus with Calculus Previews

Question

Use a graph to solve the given inequality. $$ e^{x-2}<1 $$

Step-by-Step Solution

Answer

The solution is \(x < 2\).

Step 1: 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.

Step 2: 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 \(x-2 < 0\).

Step 3: 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 \(x < 2\).

Step 4: 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.

Key Concepts

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,

• 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 \(x < 2\).

This visual method can be incredibly helpful, especially for those who find numbers and algebraic solutions abstract or difficult to visualize. A simple graph can make these concepts clear and straightforward.

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:

• 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.

Understanding these characteristics helps when solving equations and inequalities involving exponentials. By knowing how the exponential function behaves, it's easier to determine solutions just by looking at its graph.

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**:

• Recognize the form of the inequality and the function involved.
• In our case, identify that \(e^{x-2}\) should be less than 1.

2. **Simplifying the inequality**:

• Knowing the behavior of exponential functions, propose that \(x-2 < 0\) because exponential terms are less than 1 when their exponent is negative.

3. **Solving the simplified inequality**:

• Manipulate the inequality \(x-2 < 0\) to isolate \(x\).
• Add 2 to both sides, resulting in \(x < 2\).

4. **Verifying through graphical representation**:

• 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.

By following these steps systematically, you solve not just the inequality but gain insight into how each element contributes to finding a solution.