Problem 14 - Algebra Form and Function

Question

Each of the inequalities can be solved by performing two operations on both sides. State the operations in order of use and solve the inequality. $$ -2(n-3)>12 $$

Step-by-Step Solution

Answer

Question: Solve the inequality and find the range of possible values for n: -2(n-3) > 12 Answer: The solution to the inequality is n < -3, meaning that n can be any value less than -3.

Step 1: Remove the multiplier of the parentheses

We have the inequality: \(–2(n-3) > 12\). To remove the multiplier -2, we multiply both sides by a negative 1 to make it positive and then divide by 2. Note that when we multipy or divide by a negative number, we have to change the direction of the inequality. $$ \frac{-1(-2(n-3))}{2} < \frac{-1(12)}{2} $$

Step 2: Simplify the inequality

Now, simplify both sides of the inequality and solve for n. $$ (n-3) < -6 $$ Add 3 to both sides to isolate n. $$ n < -6 + 3 \Rightarrow n < -3 $$ The solution to the inequality is \(n<-3\).

Key Concepts

Understanding Inequality Operations

Inequality operations help us manipulate inequalities to find solutions. When solving inequalities like \(-2(n-3) > 12\), it's crucial to perform the same operations on both sides. This ensures the inequality remains balanced. Always remember:

• Addition/subtraction: You can add or subtract the same number from both sides without altering the inequality's direction.
• Multiplication/division: Multiplying or dividing by a positive number keeps the direction intact.
• Major caution when dealing with negative numbers, which we’ll explore further.

Starting with these basic operations, you can solve more complex inequalities by simplifying them one step at a time.

Impact of Negative Number Multiplication

Multiplying or dividing by a negative number in an inequality has a unique effect. Usually, when we multiply or divide an equation by a number, its equality remains unchanged. However, inequalities flip this logic when a negative is involved. Consider multiplying \(-2(n-3) > 12\) by \(-1\), as initially shown in the original step-by-step solution:\[\frac{-1(-2(n-3))}{2} < \frac{-1(12)}{2}\]Notice that the inequality switched from \(>\) to \(<\). This direction change is fundamental when solving inequalities. Think of it as reversing the perspective of the inequality when dealing with negatives. It's an essential skill that helps prevent errors and ensures accurate solutions.

Why Reverse Inequality Direction?

Reversing the inequality's direction when multiplying or dividing by a negative number can confuse new learners. It stems from the fundamental nature of negative numbers and how they relate on the number line. Imagine if less than zero were treated like greater than by switching the viewpoint. This mathematical rule helps maintain consistency and accuracy across number lines:

• If you multiply or divide both sides by a negative number, flip the inequality sign.
• This reversal maintains the truth of the mathematical inequality.

It's a crucial rule to remember and practice regularly as it can easily be overlooked or forgotten. Consistency in remembering to reverse the inequality direction will save you from inaccuracies in your calculations.

Working with Algebraic Expressions

Algebraic expressions often appear in inequality problems, and understanding them is essential to finding solutions. Consider the expression \(-2(n-3)\) in our inequality problem:

• Expressions combine variables, numbers, and operations.
• To simplify, remove the parentheses by applying multiplication or division as needed.

Algebra allows isolating the variable of focus by simplifying, distributing, or rearranging structures in an inequality. In our case, resolving the expression \(n-3\) involved simple arithmetic to isolate \(n\). Simplification and manipulation of expressions are central to solving inequalities.