Problem 29 - Algebra 2

Question

Define a variable and write an inequality for each problem. Then solve. The sum of a number and 8 is more than 2 .

Step-by-Step Solution

Answer

The number is greater than -6 (i.e., \(x > -6\)).

Step 1: Define the Variable

Let the variable \( x \) represent the unknown number. This is the number we are trying to find in the problem.

Step 2: Write the Inequality

According to the problem statement, "The sum of a number and 8 is more than 2" can be translated to an inequality. We express this as: \( x + 8 > 2 \).

Step 3: Solve the Inequality

To solve for \( x \), we need to isolate \( x \) on one side of the inequality. We do this by subtracting 8 from both sides: \( x + 8 - 8 > 2 - 8 \). Simplifying both sides, we get \( x > -6 \).

Step 4: Interpret the Solution

The inequality \( x > -6 \) means that the number we are looking for is any number greater than \( -6 \). This solution tells us the range of possible values for \( x \).

Key Concepts

Defining Variables

In many math problems, especially those involving inequalities, the first step is to define your variable. A variable is typically a letter that stands in for an unknown number. In our example, we are trying to find an unknown number that, when added to 8, results in a number greater than 2. We chose the letter \( x \) to represent this unknown number.

Defining the variable clearly at the beginning helps keep the entire solution organized and easy to follow. It serves as a placeholder for the value we want to find and simplifies the rest of the process.

Translating the Inequality

The next step is to translate the verbal statement into a mathematical inequality. This is often where confusion can arise, but it's simply a matter of understanding the language of math.

For our example, the problem mentions "the sum of a number and 8 is more than 2." Let's break it down:

• "The sum of a number and 8" implies we are adding 8 to the unknown number \( x \).
• "Is more than" suggests an inequality where the sum is greater than 2.

Combining these parts results in the inequality \( x + 8 > 2 \). This translation converts a word problem into a math problem we can solve.

Solving the Inequality

Once you have a mathematical inequality, the goal is to solve it. Solving inequalities often involves similar steps as solving equations, but you have to be mindful of the inequality sign.

For \( x + 8 > 2 \), we need to isolate \( x \) to find its possible values. We do this by subtracting 8 from both sides:

• Subtracting 8 from the left: \( x + 8 - 8 = x \).
• Subtracting 8 from the right: \( 2 - 8 = -6 \).

This simplifies our inequality to \( x > -6 \). The process is straightforward, but remember: if you multiply or divide by a negative number, the inequality sign flips direction, though here that step is not necessary.

Interpreting the Solution

The final step is interpreting the solution, which tells us what values \( x \) can take. From the inequality \( x > -6 \), we determine that \( x \) represents any number greater than \(-6\).

This solution gives us a range rather than a specific number and is essential for understanding the broader context of the problem:

• \( x \) could be \(-5, 0, 10, 100\), or any other number larger than \(-6\).

Understanding this range is key in many real-world scenarios where a problem may have multiple possible solutions. Proper interpretation helps apply this solution effectively.