Problem 17 - Prealgebra

Question

Solve each inequality. Check your answer. $$25

Step-by-Step Solution

Answer

The solution is \( n > 37 \).

Step 1: Isolate the Variable

We start with the inequality: \( 25 < n - 12 \). To isolate \( n \), we need to get rid of \(-12\) by adding \(12\) to both sides of the inequality. This gives us: \[ 25 + 12 < n \].Simplifying the left side, we find:\[ 37 < n \].Thus, \( n > 37 \).

Step 2: Verify the Solution

To verify that our solution \( n > 37 \) is correct, we can choose a value for \( n \) greater than \(37\), such as \(n = 38\). Substitute back into the original inequality to check:\[ 25 < 38 - 12 \].This simplifies to \( 25 < 26 \), which is true. Therefore, \( n = 38 \) is a valid solution, confirming that \( n > 37 \) is correct.

Key Concepts

Isolating the Variable

In prealgebra, solving inequalities involves a process very similar to solving equations, where the goal is to isolate the variable on one side of the inequality. Let's look at how this is done using the example: \( 25 < n + (-12) \) To start, we simplify the expression to: \( 25 < n - 12 \). The next step is to isolate \( n \). This means we need to remove \(-12\) from the right side of the inequality. We achieve this by performing the inverse operation, which in this case is adding \(12\) to both sides:

• Add \(12\) to the left side: \(25 + 12\)
• Add \(12\) to the right side: \(n - 12 + 12\)

This simplifies to: \(37 < n\). We have successfully isolated the variable \(n\). This tells us that \(n\) must be greater than \(37\). Remember the golden rule: whatever operation you do to one side, do the same to the other side. This keeps the inequality balanced.

Solution Verification

After isolating the variable, we must verify that our solution is correct. Verification ensures that the solution accurately satisfies the original inequality. Using our inequality solution \( n > 37 \), choose a value for \( n \) that meets this condition. A good choice is \( n = 38 \). Substitute this value back into the original inequality: \( 25 < 38 - 12 \) This simplifies to \( 25 < 26 \). Since this statement is true, our solution is verified. Verification is crucial, as it confirms our steps were correct and the inequality holds true for the range of values found. Selecting different values within the range can further solidify the accuracy.

Understanding Prealgebra

Prealgebra is the foundation of mathematical logic, and learning how to solve inequalities is a key skill. Inequalities communicate how quantities relate to each other in terms of size. In the provided exercise, understanding the notation \( < \) and \( > \) is essential. These symbols are used to indicate 'less than' and 'greater than,' which help to define the relationship between quantities. Inequalities differ from equations in that they allow for a range of possible solutions. Here, \( n > 37 \) means any value larger than 37 satisfies the inequality. Remember these important concepts in prealgebra:

• Inequality symbols indicate a comparison between expressions.
• Isolating the variable ensures we solve for a range of values that make the inequality true.
• Verification is a method to double-check and ensure the accuracy of the given solution.

Through these exercises, students strengthen their problem-solving skills and enhance their understanding of mathematical logic, setting a strong base for more complex math topics.