Problem 60 - 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) \(32^{x}=16^{1-x}\) (b) \(32^{x}>16^{1-x}\) (c) \(32^{x}<16^{1-x}\)

Step-by-Step Solution

Answer

(a) \(x = \frac{4}{9}\); (b) \(x > \frac{4}{9}\); (c) \(x < \frac{4}{9}\).

Step 1: Express with Common Base

Rewrite both sides of the equation \(32^{x} = 16^{1-x}\) using the same base: \(32\) and \(16\) can be expressed as powers of \(2\). So, \(32 = 2^5\) and \(16 = 2^4\). Substitute these into the equation to get: \( (2^5)^x = (2^4)^{1-x} \).

Step 2: Simplify the Equation

Apply the power rule \((a^m)^n = a^{mn}\) to both sides to get: \(2^{5x} = 2^{4(1-x)}\).

Step 3: Set the Exponents Equal

Since the bases are equal, set the exponents equal to solve for \(x\): \(5x = 4(1-x)\).

Step 4: Solve for x

Distribute and combine like terms: \(5x = 4 - 4x\).Add \(4x\) to both sides: \(9x = 4\).Divide both sides by \(9\): \(x = \frac{4}{9}\).

Step 5: Graphing the Functions

Graph the equations \(y_1 = 32^x\) and \(y_2 = 16^{1-x}\) using a calculator. The point where the graphs intersect represents the solution \(x = \frac{4}{9}\).

Step 6: Solve Inequality for Part (b)

To solve \(32^x > 16^{1-x}\), look at the graph where \(y_1\) is above \(y_2\). This happens when \(x > \frac{4}{9}\).

Step 7: Solve Inequality for Part (c)

To solve \(32^x < 16^{1-x}\), look at the graph where \(y_1\) is below \(y_2\). This occurs when \(x < \frac{4}{9}\).

Key Concepts

Understanding Inequalities

Inequalities describe a range of values rather than a specific one. In this exercise, we are comparing the expressions \(32^x\) and \(16^{1-x}\).
They help us determine which expression is greater for certain values of \(x\). Here's how we approach them:

• Greater Than (>): For \(32^x > 16^{1-x}\), we find where the value of \(32^x\) is higher than \(16^{1-x}\). Solving using our graph shows that \(x > \frac{4}{9}\).
• Less Than (<): For \(32^x < 16^{1-x}\), this inequality indicates where \(16^{1-x}\) exceeds \(32^x\), occurring when \(x < \frac{4}{9}\).

Graphing these expressions helps visualize these inequalities easily. The points where the graph of \(y_1\) is above or below \(y_2\) are critical for understanding which inequality is satisfied.

Graphing Exponential Functions

Graphing is a powerful tool for visualizing solutions to equations and inequalities. It allows us to see where different functions intersect or diverge.
For the functions \(y_1 = 32^x\) and \(y_2 = 16^{1-x}\), graphing helps in understanding the relationship between these expressions.

• Intersection Point: The graphs intersect at \(x = \frac{4}{9}\). This shows where the expressions are equal, which is the solution to \(32^x = 16^{1-x}\).
• Above and Below: By observing the graph, you can easily determine sections where \(y_1\) is greater than or less than \(y_2\). This visually complements the analytical solution for inequalities.

Using a graphing calculator can simplify this process, providing a clear picture of function behavior across different values of \(x\).

Using a Common Base

Working with a common base in exponential equations can simplify the solving process significantly. In this example, both \(32\) and \(16\) are rewritten in terms of \(2\):

• Base Conversion: Express \(32\) as \(2^5\) and \(16\) as \(2^4\).
• Rewriting the Equation: Substitute these values into \(32^x = 16^{1-x}\) to get \((2^5)^x = (2^4)^{1-x}\).
• Simplification: Applying the power rule \((a^m)^n = a^{mn}\), we simplify to \(2^{5x} = 2^{4 - 4x}\).
• Setting Equal Exponents: Since the bases are the same, solve \(5x = 4(1-x)\) to find \(x = \frac{4}{9}\).

This method of converting to a common base makes it easier to compare exponents directly and understand the relationship between exponential expressions. It's an efficient way to tackle complex exponential equations.