Problem 43 - Algebra 2

Question

Solve each equation or inequality. \(4 e^{x}-3>-1\)

Step-by-Step Solution

Answer

The solution is \(x > -\ln(2)\) or approximately \(x > -0.693\).

Step 1: Isolate the exponential term

Start by isolating the exponential term. Add 3 to both sides of the inequality:\[4e^x - 3 + 3 > -1 + 3\]This simplifies to:\[4e^x > 2\]

Step 2: Divide by coefficient

Next, divide both sides of the inequality by 4 to further isolate the exponential term:\[\frac{4e^x}{4} > \frac{2}{4}\]This simplifies to:\[e^x > \frac{1}{2}\]

Step 3: Solve for x using logarithms

To solve for \(x\), take the natural logarithm (ln) on both sides of the inequality:\[\ln(e^x) > \ln\left(\frac{1}{2}\right)\]Using the property \(\ln(e^x) = x\), we get:\[x > \ln\left(\frac{1}{2}\right)\]This further simplifies to:\[x > -\ln(2)\]

Step 4: Simplify the solution

Recognize that \(-\ln(2)\) is a constant, approximately equal to \(-0.693\). Therefore, the inequality can be expressed as:\[x > -0.693\]

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a logarithm with the base \( e \). The number \( e \) is an irrational constant approximately equal to 2.71828. It is a fundamental constant in mathematics, similar to \( \pi \).

The natural logarithm of a number is the power to which \( e \) must be raised to equal that number. For example, \( \ln(e) = 1 \) because \( e^1 = e \). Thus, the natural logarithm can be particularly useful when working with exponential functions. This property is often used to solve exponential equations or inequalities.

In the solution process of our exercise, applying the natural logarithm helps us to isolate \( x \) in the inequality \( e^x > \frac{1}{2} \). We took the natural logarithm on both sides of the inequality. Using the property \( \ln(e^x) = x \), we make the equation much simpler to solve.

Steps to Solving Inequalities

Solving inequalities involves finding the set of all possible values that make the inequality true. It's similar to solving equations but with some unique properties and steps. Here is how we solve the inequality in the exercise:

• First, we isolate the exponential expression. This involves basic algebraic manipulations like addition, subtraction, multiplication, or division.
• Next, if there's a coefficient with the exponential term, we divide both sides of the inequality to solve for the exponential expression itself.
• Finally, we apply logarithmic operations, such as the natural logarithm, to solve for the variable. While taking logarithms, maintain the inequality's nature; when the logarithm of both sides is taken, the inequality's direction remains unchanged if both sides are positive.

These steps are systematic, allowing us to logically and safely handle inequalities in mathematical problems.

Explaining Exponential Functions

An exponential function is one where the variable appears in the exponent. In simpler terms, the situation involves expressions like \( e^x \) where \( e \) is the base of the natural logarithm. Exponential functions are prevalent in describing growth and decay processes, such as population growth and radioactive decay.

In our exercise, we worked with the exponential inequality \( 4e^x - 3>-1 \). This is a type of inequality where solving involves isolating the term containing the exponential function, \( e^x \), before applying further methods such as logarithms.

Exponential functions play a crucial role because property rules like "exponents of a base \( e \)" and calculating natural logs make it relatively easier to handle changes in variables. In particular, they highlight the rapid growth or decay typical of processes modeled by these functions. Understanding and manipulating these properties are essential skills in solving many real-world problems.