Problem 123 - Introductory and Intermediate Algebra for College Students 4th

Question

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$2^{x+1}=8$$

Step-by-Step Solution

Answer

The solution to the equation \(2^{x+1}=8\) is \(x = 2\).

Step 1: Break Down the Equation

First, we need to simplify the equation. Break down \(2^{x+1}\) to \(2 * 2^x\). This makes it easier to analyse and compare both sides of our equation \(2 * 2^x = 8\).

Step 2: Graph Each Side of the Equation

Next, input the two functions, \(y = 2 * 2^x\) and \(y = 8\) into the graphing utility and plot them on the same graph. Look for the x-coordinate of the intersection point which represents the solution set.

Step 3: Find the Intersection Point

The x-coordinate of the intersection point will give the solution to the equation. Visually, this is the value of \(x\) where \(2 * 2^x = 8\). Using the graph, calculate the intersection point, which results in \(x = 2\).

Step 4: Verify using Direct Substitution

Substitute \(x = 2\) into the original equation, which gives us \(2^{2+1} = 8\). Performing the calculation, \(2^3 = 8\) indeed equals to 8, which verifies our solution.