Question

Solve the inequality.\(9^x>54\)

Short answer

The solution to the inequality is \(x > \log_{9}{6} + 1\)

Step-by-step solution

Simplify the right-hand side

We divide both sides of the inequality by 9 to simplify it: \(9^{x-1} > 6\)

Convert to logarithmic form

In this step, we convert the inequality to its logarithmic form which gives us: \(x-1 > \log_{9}{6}\).

Solve for x

In this step, we solve the inequality for x: \(x > \log_{9}{6} + 1\).

Key concepts

Exponential Inequalities

Exponential inequalities are a type of inequality where the unknown variable appears in the exponent. These can be intimidating at first, but breaking them down step by step makes them manageable.

In the exercise you provided, we have an exponential inequality: \(9^x > 54\). Here, the base of the exponent is 9. To solve it, we want to compare the expression \(9^x\) to another number on the right.

A practical method for solving such inequalities is to simplify the expression where possible. In this instance, dividing both sides of the inequality by 9 simplifies our expression to \(9^{x-1} > 6\). This step reduces the size of the exponent and makes the subsequent steps easier to handle. Remember, manipulating the inequality sign is similar to an equation, but you must consider the properties of real numbers when dividing by negative terms in inequality scenarios—not the case here since we're dealing with positive numbers. With inequalities, these simplifications often involve breaking the expression into smaller, more recognizable components.

Logarithmic Conversions

Logarithmic conversions are crucial in transforming complex exponential inequalities into simpler forms. Logarithms essentially "unwrap" the power, allowing for straightforward algebraic manipulation.

After simplifying the initial inequality, the next step is to convert it into its logarithmic form. For the given problem, the inequality \(9^{x-1} > 6\) becomes \(x-1 > \log_{9}{6}\) using logarithms. This conversion is key because it transforms the problem into a linear inequality, which is much simpler to solve with traditional algebraic methods.

Here are a few basic rules which help in these conversions:

• If \(a^b = c\), then \(b = \log_a{c}\).
• Make sure the base of the logarithm corresponds with the base of the exponential term for consistency and accuracy.
• Logarithms allow you to take advantage of their properties, such as the power rule, product rule, and change of base rule, especially if the base is not readily accessible in calculators.

Converting to logarithmic form reduces potential errors and simplifies solving steps. This procedure turns an exponential problem into one easily dealt with using linear algebra techniques.

Mathematical Problem Solving

Mathematical problem solving is often about identifying the underlying structure of a problem and applying appropriate methods to find a solution. This exercise contains exponential and logarithmic transformations, which require strategic problem-solving skills.

When solving for \(x\) in the inequality \(x-1 > \log_{9}{6}\), you want to isolate \(x\). Here, this involves adding 1 to both sides of the inequality, resulting in \(x > \log_{9}{6} + 1\).

Problem solving in mathematics often follows these key strategies:

• **Understand the problem**: Read the inequality carefully to comprehend what is being asked.
• **Plan the solution**: Determine which mathematical concepts apply. For this exercise, exponential and logarithmic considerations are central.
• **Execute the plan**: Simplify the inequality and convert where necessary. Solve the inequality using basic algebraic principles.
• **Review/Verify**: Check your result to see if it makes sense. In this problem, ensure \(x\) satisfies the original inequality \(9^x > 54\).

Through practice, these strategies become more intuitive, but understanding each step helps ensure accuracy and deepens comprehension.