Problem 75 - College Algebra

Question

In Exercises \(75-82,\) 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 \(x = 2\), as confirmed by direct substitution into the equation.

Step 1: Graph each side of the equation

On a graphing utility, plot the function \(y=2^{x+1}\) as well as the line \(y=8\). These graphs will intersect at the solution to the given equation.

Step 2: Find the intersection point

The x-coordinate of the intersection point, which can be detected from the graph, is the solution to this equation. On the 'y = 8' line, identify and note down this x-coordinate.

Step 3: Confirm the solution by direct substitution

Substitute the found x-coordinate value into the original equation \(2^{x+1}=8\), replacing \(x\) to confirm whether it validates the equation. If it does, that means the solution is correct.

Key Concepts

Graphing Utility

A graphing utility is an invaluable tool when solving exponential equations. It allows you to visually understand how the functions behave and interact. When you graph each side of the equation separately, you can see where they meet, which reveals the solution to the equation.

For the equation \(2^{x+1}=8\), this means that you will plot two graphs:

• One graph for the exponential function \(y=2^{x+1}\).
• Another graph for the horizontal line \(y=8\).

Ensure both graphs fit into the same viewing rectangle to clearly identify their intersection. This setup will help visualize where these two functions equal each other, providing a straightforward approach to solving the equation.

Intersection Point

Finding the intersection point is a crucial step when solving equations graphically. This point represents where the values of both functions coincide. On the graph you've plotted using a graphing utility, look carefully for where the two curves cross.

In our specific example with \(y=2^{x+1}\) and \(y=8\), the intersection point will occur at a particular \(x\) value. The \(x\)-coordinate of this intersection tells us the solution to the equation. In practical terms:

• Find the spot where the curve \(y=2^{x+1}\) meets the line \(y=8\).
• Record the \(x\) coordinate at this intersection point.

Once you have this value, it is essential to make sure this is indeed the correct solution by further verification.

Direct Substitution

After identifying the \(x\)-coordinate from the intersection point, it's time to verify your solution. Direct substitution is a simple, yet powerful method to confirm its correctness.

With direct substitution, you'll replace \(x\) in the original equation with the value found from the graph. For \(2^{x+1}=8\), substituting your \(x\)-value should simplify to true equality:

• Calculate \(2^{(x + 1)}\) with your \(x\) value.
• Check if it equals 8.

If these calculations validate the original equation, you've confirmed the solution is correct. This technique double-checks your graphical findings, ensuring accuracy in your equation-solving process.