Problem 16 - Algebra 2

Question

Solve each inequality. Check your solution. $$ 3^{3 x-2}>81 $$

Step-by-Step Solution

Answer

The solution is \(x > 2\).

Step 1: Recognize the inequality form

The inequality given is \(3^{3x-2}>81\). Notice that both sides of the inequality can be expressed in terms of base 3 as follows: 81 can be written as \(3^4\).

Step 2: Rewrite 81 as a power of 3

Rewrite the given inequality by expressing 81 as \(3^4\):\[3^{3x-2} > 3^4\]

Step 3: Compare exponents

Since the bases are the same and greater than 0, the inequality \(3^{3x-2} > 3^4\) implies that:\[3x - 2 > 4\]

Step 4: Solve the linear inequality

Solve the inequality \(3x - 2 > 4\) by isolating the variable \(x\): Add 2 to both sides: \[3x > 6\] Then divide both sides by 3: \[x > 2\]

Step 5: Check the solution

Choose a value greater than 2 (e.g., \(x = 3\)) to test if the solution holds true:\(3^{3(3)-2} = 3^{7} = 2187\) which is greater than 81. Thus, the solution is correct, and \(x > 2\).

Key Concepts

Base Exponentiation

Base exponentiation is a key concept when working with expressions that involve powers of numbers. It involves expressing numbers as powers, where a "base" number is raised to an "exponent" or "power." In the inequality given, the expression on the left side can be identified as an exponentiation with base 3. Similarly, the number 81 on the right side can also be rewritten as an exponentiation with the same base.Here's how base exponentiation works:

• A base number is the number that is multiplied by itself a certain number of times.
• The exponent or power tells us how many times to use the base number in the multiplication.
• For example, with base 3 and exponent 4, we write it as \(3^4\), which equals \(3 \times 3 \times 3 \times 3 = 81\).

In this exercise, identifying that both sides of the inequality can be expressed with the same base (3) helps in comparing the exponents directly. This simplification allows us to solve the inequality more easily.

Solving Linear Inequalities

Linear inequalities are like linear equations but use inequality signs (>, <, ≥, ≤) instead of an equals sign. Solving them involves manipulating the inequality to isolate the variable. The goal is to determine the range of values that the variable can take to make the inequality true.Here’s a step-by-step approach:

• First, simplify the inequality if needed. For example, rewrite expressions using a common base like we did with \(3^{3x-2} > 3^4\).
• Once the exponents have the same base, compare them directly. Since \(3\) is positive and greater than 1, we know that \(3x - 2 > 4\).
• To find the range of \(x\), perform operations to isolate \(x\). Add 2 to each side, resulting in \(3x > 6\). Then, divide through by 3 to obtain \(x > 2\).

Remember, when dealing with inequalities:

• If you multiply or divide both sides by a negative number, reverse the inequality sign.
• Maintain the direction of the inequality sign when performing other arithmetic operations.

Solving linear inequalities is crucial as it tells us the set of all possible solutions.

Checking Solutions

Checking solutions in inequalities ensures that the derived range for the variable is correct. This step involves selecting a number from the solution set and substituting it into the original inequality to confirm that it holds true.For instance, after solving \(3x - 2 > 4\) and finding \(x > 2\), we select a number greater than 2, like \(x = 3\), and substitute it back:

• Calculate \(3^{3(3)-2} = 3^7\).
• The result is \(2187\), which is indeed greater than \(81\), thus confirming the inequality.

By verifying with actual calculations, we double-check whether all steps in our solving process hold. Checking solutions is a practical exercise that ensures our mathematical reasoning is accurate. It confirms that the solution \(x > 2\) is valid and fully satisfies the original inequality \(3^{3x-2} > 81\). This verification process is essential for robust understanding and error prevention.