Problem 57 - A Graphical Approach to College Algebra

Question

Solve each equation in part (a) analytically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(2^{x+1}=8\) (b) \(2^{x+1}>8\) (c) \(2^{x+1}<8\)

Step-by-Step Solution

Answer

(a) \(x = 2\); (b) \(x > 2\); (c) \(x < 2\).

Step 1: Solve the Equation Analytically

The equation given is \(2^{x+1} = 8\). First, rewrite 8 as a power of 2: \(8 = 2^3\). Therefore, the equation becomes \(2^{x+1} = 2^3\). Since both sides have the same base, we can set the exponents equal to each other: \(x + 1 = 3\). Solve for \(x\) by subtracting 1 from both sides: \(x = 2\).

Step 2: Graph the Equation Using a Calculator

Plot the function \(y = 2^{x+1}\) on your calculator. Identify the point where the graph intersects the horizontal line \(y = 8\). This intersection occurs at \(x = 2\), confirming our analytical solution.

Step 3: Solve the Inequality for Part (b)

The inequality for part (b) is \(2^{x+1} > 8\). Using the previous step's graph, observe where the graph of \(y = 2^{x+1}\) is above the line \(y = 8\). This occurs for values of \(x\) greater than 2. Therefore, the solution to the inequality is \(x > 2\).

Step 4: Solve the Inequality for Part (c)

The inequality for part (c) is \(2^{x+1} < 8\). Again, refer to the graph where \(y = 2^{x+1}\) is below the line \(y = 8\). This occurs for values of \(x\) less than 2. Therefore, the solution to the inequality is \(x < 2\).

Key Concepts

Graphical Solutions

Graphical solutions allow us to visualize mathematical equations and inequalities. This method involves plotting the functions on a graph and analyzing their intersections or relative positions. It helps us understand the behavior of functions over their domains in a clear, intuitive manner.
To solve the equation graphically, we consider both sides as separate functions and plot them. For example, with the equation \(2^{x+1} = 8\):

• Plot the function \(y = 2^{x+1}\).
• Draw a horizontal line where \(y = 8\) (since \(8 = 2^3\)).

The solution to the equation is the value of \(x\) at which these two graphs intersect. This intersection demonstrates the equivalence of both expressions at that point. Graphical methods are particularly helpful when looking for multiple solutions or when dealing with complex equations that are tough to solve analytically.

Inequalities

Inequalities involve finding out the range of values for which an expression holds true. They are slightly more complex than equations because they specify a range rather than a fixed point. Consider the inequalities related to \(2^{x+1}\).

• For \(2^{x+1} > 8\), we need to identify where \(y = 2^{x+1}\) is above the line \(y = 8\).
• Conversely, for \(2^{x+1} < 8\), the task is to find where \(y = 2^{x+1}\) lies below the line \(y = 8\).

These conditions divide the graph into regions. In the first case, the expression is true for values of \(x > 2\). In the second case, it's true for \(x < 2\). This graphical examination makes inequalities easier to interpret because of the visual separation of solution boundaries.

Calculator Graphing

Calculator graphing is a great tool for confirming analytical solutions and exploring equations that are challenging to solve by hand. Graphing calculators allow us to input functions and immediately observe their behavior.

• Enter the function \(y = 2^{x+1}\) into the calculator.
• Use the calculator to draw the graph, and add the line \(y = 8\).

By doing so, you can quickly see where the function crosses the reference line. This intersection confirms the analytic solution that \(x = 2\) solves \(2^{x+1} = 8\). Additionally, calculator graphing can show where the function is greater or less than the given value, thus providing a visual method to solve inequalities like \(2^{x+1} > 8\) and \(2^{x+1} < 8\). As a result, the tool offers a practical approach to verify results and explore functions with accuracy and ease.