Problem 33 - Precalculus with Calculus Previews

Question

In Problems 33 and 34 , use a graph to solve the given inequality. $$ \ln (x+1)<0 $$

Step-by-Step Solution

Answer

The solution to \( \ln(x+1) < 0 \) is \( x < 0 \).

Step 1: Understanding the Inequality

First, we need to understand what the inequality \( \ln(x+1) < 0 \) means. This inequality asks us to find all values of \( x \) such that when \( x + 1 \) is inserted into the natural logarithm function \( \ln \), the result is less than zero.

Step 2: Interpret Logarithmic Property

The inequality \( \ln(y) < 0 \) translates to \( y < 1 \) because the logarithm of any number less than 1 is negative. This stems from the basic property that \( \ln(1) = 0 \) and the natural logarithm function is continuous and increases on \((0, \infty)\).

Step 3: Setting Up the Inequality

Since we need \( y = x+1 \) to be less than 1, we set up the inequality: \( x + 1 < 1 \).

Step 4: Solving the Inequality

Now, solve the inequality for \( x \):\[ x + 1 < 1 \] To isolate \( x \), subtract 1 from both sides:\[ x < 0 \]

Step 5: Graphical Interpretation

To find the solution visually with a graph, consider the function \( y = \ln(x+1) \) and plot it. The graph of \( \ln(x+1) \) is defined for \( x > -1 \). You will notice that it crosses the \( x \)-axis at \( x = 0 \), indicating where \( \ln(x+1) = 0 \). The region where the graph is below the \( x \)-axis (\( y = 0 \)) occurs when \( x < 0 \).

Step 6: Conclusion

The solution derived from both the algebraic approach and the graphical approach is that \( x < 0 \) satisfies the given inequality \( \ln(x+1) < 0 \).

Key Concepts

Understanding Natural Logarithms

The natural logarithm, denoted as \( \ln \), is a specific kind of logarithm that uses the base \( e \). Here, \( e \) represents a constant approximately equal to 2.71828. Unlike logarithms with other bases, the natural logarithm is especially important in mathematics and science due to its natural occurrence in various growth processes.
When you apply the natural logarithm to a number, you are essentially finding the power to which \( e \) must be raised to produce that number. For example, if \( \ln(y) = 0 \), then \( e^0 = y \), which tells us that \( y = 1 \).

It is critical to remember the properties of natural logarithms when solving inequalities.

• The natural logarithm of numbers less than 1 is negative.
• \( \ln(1) = 0 \).
• \( \ln(x) \) is undefined for \( x \leq 0 \) since the logarithm is only defined for positive inputs.

These properties guide us in determining the conditions whereby the logarithmic output is less than zero.

Graphical Solution Method

A graphical solution offers a visual representation of the inequality \( \ln(x+1) < 0 \). Plotting the graph of \( y = \ln(x+1) \) can significantly assist in understanding the behavior of the function and how it relates to the inequality.
On the graph, you'll notice several critical features:

• The graph is only defined for \( x > -1 \), as \( \ln \) cannot operate on non-positive numbers.
• At \( x = 0 \), the graph crosses the x-axis because \( \ln(1) = 0 \).
• The region where the curve dips below the x-axis represents values of \( x \) that satisfy our inequality of being less than zero.

Hence, visually, you can confirm that the solution \( x < 0 \) aligns with the area where the graph is positioned below the x-axis. Graphing this function effectively corroborates the algebraic findings by translating them into a visual context.

Using an Algebraic Approach

The algebraic approach starts by reassessing the inequality \( \ln(x+1) < 0 \). The key step here is understanding that for a natural logarithm to be negative, its argument must be less than 1.
This reasoning leads us to:

• If \( \ln(y) < 0 \), then \( y < 1 \).
• Therefore, if \( y = x+1 \), then \( x+1 < 1 \).

By rearranging \( x+1 < 1 \), you simply solve for \( x \) as follows:

Subtract 1 from both sides to achieve: \[ x + 1 - 1 < 1 - 1 \]
Leading to the solution: \[ x < 0 \]

This straightforward calculation confirms the inequality's solution algebraically. Using this method, you apply basic algebraic principles to unravel the inequality without needing any graphical assistance. It shows the logical consistency across mathematics in resolving inequalities.