Problem 29 - Precalculus with Calculus Previews

Question

Use a graph to solve the given inequality. $$ 2^{x}>16 $$

Step-by-Step Solution

Answer

The solution is \(x > 4\).

Step 1: Rewrite the Inequality in Exponential Form

The inequality given is \(2^x > 16\). We know that 16 can be rewritten as a power of 2, specifically \(16 = 2^4\). So, we can rewrite the inequality as \(2^x > 2^4\).

Step 2: Set Up the Equation for Graphing

To solve using a graph, we'll graph the functions \(y_1 = 2^x\) and \(y_2 = 16\). We aim to find where \(y_1\) is greater than \(y_2\). Plot both functions on the same set of axes.

Step 3: Graph the Functions

Draw the curve for \(y_1 = 2^x\), which is an exponential curve starting from (0,1) and rising rapidly as x increases. Also, plot the horizontal line \(y_2 = 16\), which is a straight line parallel to the x-axis at \(y=16\).

Step 4: Identify the Intersection Point

Determine the x-coordinate where the curve \(y_1 = 2^x\) intersects the line \(y_2 = 16\). This occurs when \(2^x = 16\), or equivalently, when \(x = 4\).

Step 5: Analyze the Graph to Solve the Inequality

Since \(2^x\) is greater than 16 when the curve is above the horizontal line \(y = 16\), observe the area on the graph where the curve \(2^x\) is above this line. This happens for all points where \(x > 4\).

Step 6: Conclude the Solution

From the graph analysis, we conclude that the solution to \(2^x > 16\) is \(x > 4\).

Key Concepts

Graphing Techniques

Graphing is a powerful tool in mathematics that helps us visually inspect relationships between functions and solve inequalities. To graph an inequality like \(2^x > 16\), first, we need to understand the functions involved.
Begin by plotting the exponential function \(y_1 = 2^x\). This curve rises rapidly as \(x\) increases because it is an exponential function, and it starts from point \((0, 1)\) on the y-axis. The graph of \(2^x\) shows how quickly exponential growth can outpace other functions as you move along the x-axis.
Next, plot a horizontal line for the constant function \(y_2 = 16\). It forms a straight line parallel to the x-axis, with a consistent height of 16, cutting across the y-axis.
To solve the inequality, you need to determine where the curve \(2^x\) is above the line \(y=16\). Intersection points are critical to identify ranges on the graph where one function surpasses another, helping us find solutions to inequalities like \(x > 4\).
Graphs, thus, offer a visual and intuitive way of solving mathematical problems, making complex inequalities more approachable.

Exponential Functions

Exponential functions are key players in mathematics, characterized by their powerful growth rates. An exponential function can generally be expressed in the form \(f(x) = a^x\), where \(a\) is a constant base, and \(x\) is the exponent that varies.
The base here, \(a\), must always be positive and greater than 1, as negative or fractional bases introduce different behaviors not typically considered exponential in this context.
For the inequality \(2^x > 16\), we are dealing with a base of 2. The remarkable property of exponential functions is how they quickly increase as the value of \(x\) grows. Even a small increase in \(x\) dramatically increases \(2^x\).
In our specific example, knowing that \(16 = 2^4\), lets us compare directly by converting the same base. This simplifies solving the inequality because the solution depends only on the exponent: \(x > 4\).
Exponential growth is widely found in natural processes, such as population growth and radioactive decay, underscoring its importance in diverse real-world applications.

Inequalities

Inequalities are expressions that demonstrate the relative size or order of two objects. They are denoted using symbols like \(>\), \(<\), \(\geq\), and \(\leq\). In mathematics, inequalities like \(2^x > 16\) allow us to find ranges of values rather than a single solution.
Solving inequalities involves finding all values of the variable that make the inequality true. Graphically, this involves identifying where one function is greater than or less than another after plotting both on a coordinate system.
Here, with \(2^x > 16\), the inequality involves finding the \(x\) values where the plot of the exponential function \(2^x\) exceeds that of the constant line \(y = 16\).
As revealed in the graph, the \(2^x\) curve exceeds the horizontal line past the intersection point \(x = 4\). Thus, all \(x\) values greater than 4 satisfy the inequality.
Inequalities are essential in defining ranges of solutions in various mathematical fields, giving us a deeper understanding of constraints and conditions.