Problem 59 - 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) \(27^{4 x}=9^{x+1}\) (b) \(27^{4 x}>9^{x+1}\) (c) \(27^{4 x}<9^{x+1}\)

Step-by-Step Solution

Answer

(a) \(x = \frac{1}{5}\). (b) \(x > \frac{1}{5}\). (c) \(x < \frac{1}{5}\).

Step 1: Express with Common Bases

Rewrite the equation with the same base. Notice that \( 27 = 3^3 \) and \( 9 = 3^2 \). Thus, we can express the equation as \( (3^3)^{4x} = (3^2)^{x+1} \).

Step 2: Apply Power of a Power Property

Apply the power of a power property \((a^m)^n = a^{m \times n} \). So the equation becomes \(3^{12x} = 3^{2(x+1)} \).

Step 3: Equate the Exponents

Since the bases are equal, we can equate the exponents: \(12x = 2(x + 1)\).

Step 4: Solve for x

Expand and simplify: \(12x = 2x + 2\). Subtract \(2x\) from both sides: \(10x = 2\). Divide by \(10\) to isolate \(x\): \(x = \frac{1}{5}\).

Step 5: Interpret Graph for Inequality Part (b)

Graph the functions \(y_1 = 27^{4x}\) and \(y_2 = 9^{x+1}\). Look for the region where \(y_1 > y_2\). Since \(x = \frac{1}{5}\) is the point where they are equal, \(27^{4x} > 9^{x+1}\) when \(x > \frac{1}{5}\).

Step 6: Interpret Graph for Inequality Part (c)

Examine the graph to determine where \(27^{4x} < 9^{x+1}\). From the equal point \(x = \frac{1}{5}\), \(27^{4x} < 9^{x+1}\) when \(x < \frac{1}{5}\).

Key Concepts

Inequalities

Understanding inequalities is key to solving equations where values are not equal.Inequalities express a range rather than exact quantities. For example:

• The inequality \( a > b \) means "a is greater than b".
• The inequality \( a < b \) means "a is less than b".

In the context of exponential inequalities, like in parts (b) and (c) of our problem, we need to determine when one exponential expression is greater or less than another.
This involves not just solving an equation but understanding the behavior of both sides of the inequality. Inequalities involve comparing more than just values; we compare their growth rates too.
Graphing can significantly help in visualizing where one function overtakes another.

Graphical Solution

Graphical solutions provide a visual means to understand equations and inequalities. To solve an equation graphically, plot both sides of the equation as separate functions.

• For example, you're plotting \( y_1 = 27^{4x} \) and \( y_2 = 9^{x+1} \).
• Where these plots intersect corresponds to solutions where the equations are equal.

By observing these graphs, you can easily see where the functions increase relative to each other.
This helps in solving inequalities. For example, for \( 27^{4x} > 9^{x+1} \), after identifying the intersection (at \( x = \frac{1}{5} \)), look at regions to the right (where \( x > \frac{1}{5} \)) where the first function is above the second one.
Similarly, for \( 27^{4x} < 9^{x+1} \), check the region to the left of this intersection (where \( x < \frac{1}{5} \)). Graphs simplify complex equations and inequalities by translating them into visual relationships.

Power of a Power Property

The power of a power property is a fundamental concept in simplifying exponential expressions. This property states that when an exponentiated base is itself raised to another power, you can multiply the exponents: \((a^m)^n = a^{m \times n}\).
In solving the given equation \((3^3)^{4x} = (3^2)^{x+1}\), the power of a power property allows turning it into \(3^{12x} = 3^{2(x+1)}\).
Once the equation is in this standard form with like bases, we can focus only on the exponents and solve much like a linear equation.
This is a powerful technique that simplifies complex expressions and makes solving exponential equations more manageable.

• Remember to ensure bases match before applying this property, as it doesn't hold for different bases.
• This can be combined with other methods, like factoring, graphing, or using logarithms for more complex problems.

Mastering this property is essential to handling a wide variety of math problems involving exponents.