Problem 576 - Precalculus

Question

Find the exact solution for \(-5 e^{-4 x-1}-4=64 .\) If there is no solution, write no solution.

Step-by-Step Solution

Answer

No real solution exists because \(e^{-4x-1}\) cannot equal a negative value.

Step 1: Isolate the Exponential Term

First, add 4 to both sides of the equation to isolate the exponential term:\[-5e^{-4x-1} - 4 + 4 = 64 + 4\]This simplifies to:\[-5e^{-4x-1} = 68\]

Step 2: Divide to Get the Exponential Alone

Next, divide both sides of the equation by -5 to solve for the exponential expression:\[e^{-4x-1} = \frac{68}{-5}\]Simplify to:\[e^{-4x-1} = -13.6\]

Step 3: Analyze the Possibility

Recognize that the equation \(e^y = -13.6\) cannot be true for any real number \(y\) because the exponential function always yields positive values. Thus, there is no real solution.

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of algebra, allowing us to find unknown values that satisfy a given mathematical equation. The goal is to manipulate the equation to isolate the unknown variable. In this specific exercise, we dealt with an equation involving an exponential function, which requires a strategic approach for solving. Here’s a simplified process for solving equations:

• Identify the Structure: Determine the type of equation you are dealing with – linear, quadratic, or in this case, exponential.


• Isolate the Variable: Begin by performing operations that help isolate the term with the unknown variable.


• Apply Inverse Operations: Utilize inverse operations to solve for the variable, like adding if subtraction is present, or dividing to counteract multiplication.


• Reassess the Equation: Always check if the resulting solution is valid by substituting it back into the original equation.

The complexity increases with more intricate functions such as exponentials, requiring additional considerations like domain restrictions.

Exponential Functions

Exponential functions describe situations where quantities grow or decay at a constant relative rate. They are of the form \(y = a \cdot e^{bx} + c\), where \(e\) is Euler's number, approximately 2.718. Notably, these functions are continuous and defined for all real numbers, which profoundly impacts their behavior and applications.

• Positive Nature: The exponential function \(e^x\) is always positive. This means it does not produce negative values, which is crucial when determining the existence of solutions.


• Domain and Range: It is applicable for all real numbers \(x\), but provides only positive outputs, specifically in the range \((0, \infty)\).


• Applications: These functions model real-world phenomena such as population growth, radioactive decay, and interest calculations due to their rapid increase or decline.

Understanding the properties of exponential functions aids in solving equations like the one in our exercise, especially in recognizing when a real solution is impossible.

Real Numbers

Real numbers form the backbone of most basic mathematics, encompassing all numbers that can be found on the number line. They include integers, rational numbers (fractions), and irrational numbers. Understanding their scope is crucial when solving equations, particularly in deciding solution availability.

Properties of Real Numbers

- Complete Set: The set of real numbers is complete, meaning any number between two real numbers is also a real number.
- No Infinite Values: Real numbers do not include infinity or imaginary numbers; they are finite and quantifiable.

In the Context of Exponentials

For exponential equations, it's important to recognize that while exponentials have the real number domain, their range excludes negative numbers. This exclusion was evident when our equation \(e^{-4x-1} = -13.6\) proved nonexistent in the realm of real numbers, affirming no real solution exists for such cases.