Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation in part (a) analyrically. 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: Simplify the Equation

First, let's express both sides of the equation in terms of powers of 2. We know that \(32 = 2^5\) and \(16 = 2^4\). Therefore, we can rewrite the equation \(32^x = 16^{1-x}\) as \((2^5)^x = (2^4)^{1-x}\).

Step 2: Apply Exponent Rules

Using the exponent rule \((a^m)^n = a^{m \cdot n}\), we simplify both sides to get \(2^{5x} = 2^{4(1-x)}\).

Step 3: Set Exponents Equal

Since the bases are the same (both 2), we can set the exponents equal to each other: \(5x = 4(1-x)\).

Step 4: Solve for x

Expand the right side: \(4 \cdot (1-x) = 4 - 4x\). The equation now reads \(5x = 4 - 4x\). Add \(4x\) to both sides to gather like terms: \(5x + 4x = 4\). This simplifies to \(9x = 4\). Finally, divide both sides by 9 to solve for \(x\): \(x = \frac{4}{9}\).

Step 5: Verify with Calculator Graph

Use a graphing calculator to graph \(y_1 = 32^x\) and \(y_2 = 16^{1-x}\). Check the intersection point, which should confirm that \(x = \frac{4}{9}\).

Step 6: Solve Inequality (b)

For \(32^x > 16^{1-x}\), note that this inequality represents values of \(x\) where the graph of \(32^x\) lies above the graph of \(16^{1-x}\). From the graph, observe that this happens for \(x > \frac{4}{9}\).

Step 7: Solve Inequality (c)

For \(32^x < 16^{1-x}\), identify the values of \(x\) where the graph of \(32^x\) lies below the graph of \(16^{1-x}\). This occurs for \(x < \frac{4}{9}\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In the context of the given exercise, we deal with base numbers 32 and 16. Both can be expressed in terms of another base, 2, because:

• 32 is equivalent to \(2^5\)
• 16 is equivalent to \(2^4\)

This is helpful because it allows us to rewrite complicated exponentials in simpler terms. By expressing each value as a power of 2, the problem becomes much easier to solve.
Exponent rules, such as \((a^m)^n = a^{m \cdot n}\), play a vital role in simplifying problems. It is crucial to remember that if the bases are the same, you can set the exponents equal to each other. This principle was used to solve the equation \(32^x = 16^{1-x}\) by equating the expressions \(5x = 4(1-x)\). Understanding these rules is essential to handling exponential equations effectively.

Inequalities

Inequalities are used to compare two expressions and describe the range over which one expression is greater than or less than another. In the exercise, after finding the equality solution \(x = \frac{4}{9}\), the task was to solve inequalities:

• \(32^x > 16^{1-x}\)
• \(32^x < 16^{1-x}\)

These inequalities are interpreted graphically using the graphs of \(32^x\) and \(16^{1-x}\). In simpler terms:

• The inequality \(32^x > 16^{1-x}\) means identifying values where the graph of \(32^x\) is positioned above \(16^{1-x}\).
• Conversely, \(32^x < 16^{1-x}\) means finding where \(32^x\) is below \(16^{1-x}\).

Inequalities are crucial, as they reveal the interactions between different algebraic expressions over ranges of values, rather than fixed points alone.

Graphical Analysis

Graphical analysis involves using graphs to visualize mathematical functions and solve related problems. In this exercise, after solving the equation analytically, a graphing calculator was used to graph both functions, \(y_1 = 32^x\) and \(y_2 = 16^{1-x}\).
By plotting these functions, you can visually verify the solution \(x = \frac{4}{9}\) at their intersection point. This visual confirmation is helpful, especially when equations are complex, because it provides a clear picture of where two expressions are equal.
Additionally, using graphical analysis, inequalities can be checked. Observing where one curve lies above or below another directly illustrates the inequalities:

• Intersection point shows \(x = \frac{4}{9}\).
• For \(x > \frac{4}{9}\), the graph \(32^x\) is above \(16^{1-x}\); hence, \(32^x > 16^{1-x}\).
• For \(x < \frac{4}{9}\), it is below; thus, \(32^x < 16^{1-x}\).

Graphical interpretation is an excellent tool for confirming analytical work and providing deeper insight into how functions behave in various conditions.