Problem 45 - Algebra 2

Question

Solve each inequality. Check your solution. $$ 3^{n-2}>27 $$

Step-by-Step Solution

Answer

The solution is \(n > 5\).

Step 1: Express 27 as a Power of 3

The goal is to express the number 27 as a power of 3. Since \(27 = 3^3\), the inequality \(3^{n-2} > 27\) becomes \(3^{n-2} > 3^3\).

Step 2: Simplify the Exponent Inequality

Since the bases are the same (both are 3), we can simplify the inequality by comparing the exponents: \(n-2 > 3\).

Step 3: Solve for n

Solve the equation \(n-2 > 3\) by adding 2 to both sides of the inequality: \(\begin{align*}n-2 &> 3 &> 5\end{align*}\)

Step 4: Check the Solution

Choose a value for \(n\) that is greater than 5 and plug it back into the original inequality to verify. For example, let \(n = 6\). Then \(3^{6-2} = 3^4 = 81\) and since \(81 > 27\), our solution is verified.

Key Concepts

Exponent Rules

Exponents represent repeated multiplication. For example, when we see a number like \(3^4\), it means we are multiplying 3 by itself four times: \(3 \times 3 \times 3 \times 3\). Understanding exponent rules can simplify our mathematical tasks. There are several key exponent rules:

• **Product of Powers Rule**: \(a^m \times a^n = a^{m+n}\).
• **Quotient of Powers Rule**: \( \frac{a^m}{a^n} = a^{m-n} \) when \(a eq 0\).
• **Power of a Power Rule**: \((a^m)^n = a^{m \cdot n}\).
• **Zero Exponent Rule**: \(a^0 = 1\) when \(a eq 0\).
• **Negative Exponent Rule**: \(a^{-n} = \frac{1}{a^n}\).

In the original exercise, we applied these rules by expressing 27 as \(3^3\). This step allows us to work with the same base and compare the exponents directly, making the inequality easier to solve.

Solving Inequalities

Solving inequalities is similar to solving equations, but with one key difference: inequalities express a relationship of greater than or less than between expressions. Here’s how you solve a simple inequality:

• **Isolate the variable**: Just like in an equation, you often start by isolating the variable on one side.
• **Perform the same operation on both sides**: You can add, subtract, multiply, or divide both sides of the inequality by the same number (just be careful with multiplication or division by negative numbers; this reverses the inequality symbol).

In our example, the inequality \(3^{n-2} > 3^3\) simplifies to \(n-2 > 3\) by comparing exponents directly. We then add 2 to both sides, resulting in \(n > 5\). This tells us that any number greater than 5 will satisfy the inequality.

Checking Solutions

After solving an inequality, it’s crucial to check your solution to ensure it’s correct. Verification helps to confirm the result is accurate, and it follows these steps:

• **Choose a test value**: Select a number greater than the found solution. In the example, if \(n > 5\), take \(n = 6\) for verification.
• **Substitute back into the original inequality**: Substitute the chosen test value back into the original inequality to see if the statement holds true.
• **Evaluate the result**: If the inequality holds true with your test value, your solution is likely correct.

In the example provided, substituting \(n = 6\) into the original inequality results in \(3^{4} = 81\), which is indeed greater than 27. Therefore, the solution \(n > 5\) is verified to be correct.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operation symbols that represent a mathematical relationship. They are a key part of many types of math problems, including inequalities like the one in our example. Here’s a breakdown of key concepts:

• **Variables**: Symbols used to represent unknown values. In our example, \(n\) is the variable.
• **Constants**: Known numbers in the expression, like 3 and 27 in the inequality.
• **Operations**: Include addition, subtraction, multiplication, and division.
  Understanding these components helps in creating, simplifying, and solving expressions effectively.

In our inequality \(3^{n-2} > 27\), you see these components working together, with the exponent \(n-2\) being a simple algebraic expression. It’s important to handle algebraic expressions carefully to maintain the correct relationships as expressed in the inequalities.