Problem 62 - Algebra 2

Question

Solve each equation or inequality. Check your solution. \(3^{x-2} \geq 27\)

Step-by-Step Solution

Answer

The solution is \(x \geq 5\).

Step 1: Express 27 as a Power of 3

Recognize that 27 can be expressed as a power of 3. Since \(27 = 3^3\), we can rewrite the inequality as \(3^{x-2} \geq 3^3\).

Step 2: Set Up the Exponent Inequality

Since the bases are the same and they are greater than zero, we can equate the exponents for the inequality. This gives us \(x - 2 \geq 3\).

Step 3: Solve the Inequality for x

Add 2 to both sides of the inequality to isolate \(x\). This gives you: \(x \geq 3 + 2\), which simplifies to \(x \geq 5\).

Step 4: Check the Solution

To verify, substitute \(x = 5\) back into the original inequality: \(3^{5-2} \geq 27\), which simplifies to \(3^3 \geq 27\), and this is true since both sides are equal. Hence, solutions \(x \geq 5\) satisfy the inequality.

Key Concepts

Exponential Equations

Exponential equations are types of equations where variables appear as exponents. They play a critical role in algebra and are often used to model growth and decay processes. To solve exponential equations, the key is to express the numbers involved in terms of the same base. For example, when dealing with the equation \(3^{x-2} \geq 27\), notice how the number 27 can be rewritten using base 3 as \(3^3\). This allows us to compare the exponents directly, turning the exponential equation into a simpler linear form. The rule here is that if the bases are identical, then their exponents can be equated or compared directly depending on the inequality or equality sign used. This strategy greatly simplifies the solution process and helps students build a solid foundation for more complex algebraic concepts.

Checking Solutions

Checking solutions is an essential step in solving equations and inequalities. Why? Because it ensures that the solution is correct and fully satisfies the original problem. After finding a potential solution, always plug it back into the original equation or inequality. For instance, once we found \(x \geq 5\) as a solution to the inequality \(3^{x-2} \geq 27\), we substitute \(x = 5\) into the original inequality. This gives us \(3^{3} \geq 27\), which simplifies to \(27 \geq 27\), a true statement. This confirms that our solution is indeed correct. This practice not only helps in verifying correctness but also hones the problem-solving skills by challenging assumptions and ensuring precision in results.

Inequality Properties

Inequality properties are the rules that govern how inequalities function. They dictate how we manipulate such expressions while maintaining their truth. For example, when solving \(3^{x-2} \geq 3^3\), we used the property that when both sides of an inequality have the same positive base, the exponents can be directly compared. This leads to a new inequality \(x-2 \geq 3\). Moreover, standard inequality manipulations apply, such as adding or subtracting the same number from both sides. This rule helped us isolate \(x\) as we added 2 to both sides, ultimately arriving at \(x \geq 5\). Such properties are crucial because they maintain the integrity of inequalities while transforming them into more solvable forms. Understanding and applying these properties correctly is vital for mastering mathematical inequalities.