Problem 124 - 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. $$3^{x+1}=9$$

Step-by-Step Solution

Answer

The solution set to the equation \(3^{x+1}=9\) is \(x=1\).

Step 1: Identify the equation and constant

The equation given is \(3^{x+1}\) and the constant is \(9\). It's required to graph these separately on a graphing utility.

Step 2: Graph and find the intersection

Using a graphing utility, graph \(y=3^{x+1}\) and \(y=9\) on the same graph. The point where both graphs intersect is the solution to the equation. Record the \(x\)-coordinate of this intersection.

Step 3: Verification via Direct Substitution

Take the \(x\)-coordinate from step 2, and plug it back into the equation. If the right-hand side equals 9 after substitution, then the \(x\)-coordinate solution is correct.

Step 4: Simplify the equation

Another way to solve the equation is by translating it to a simpler form. The equation \(3^{x+1}=9\) is equivalent to \(3^{x+1} = 3^{2}\), which implies that \(x+1=2\). Solving for \(x\), we find \(x=2-1=1\).

Step 5: Verification via Direct Substitution 2

Substitute \(x = 1\) back into the original equation, \(3^{x+1}=9\). If it equals 9, then the solution is correct.