Problem 4 - Prealgebra

Question

Solve each inequality. Check your solution. $$a-5>6$$

Step-by-Step Solution

Answer

The solution is \( a > 11 \).

Step 1: Set Up the Inequality

We start with the inequality given in the problem: \[ a - 5 > 6 \] This is an inequality involving a single variable, \( a \). Our goal is to solve for \( a \), which means finding the set of values that \( a \) can take to satisfy this inequality.

Step 2: Isolate the Variable

To solve for \( a \), we need to isolate it on one side of the inequality. To do this, we add 5 to both sides:\[ a - 5 + 5 > 6 + 5 \]This simplifies to:\[ a > 11 \]Now, the inequality has \( a \) by itself, and we have solved the inequality.

Step 3: Check the Solution

Choose a value for \( a \) that is greater than 11 to verify that our solution is correct. Let's try \( a = 12 \): Substitute \( a = 12 \) back into the original inequality:\[ 12 - 5 > 6 \]Simplifying gives:\[ 7 > 6 \]Since this statement is true, our solution \( a > 11 \) is correct.

Key Concepts

Variable Isolation

In the context of solving inequalities, variable isolation is a key step that aims to get the variable by itself on one side of the inequality sign. This is crucial because it helps us clearly understand the possible values that the variable can take.
The main strategy in variable isolation is to perform arithmetic operations that will eliminate any constants or coefficients attached to the variable. For instance, if we encounter an inequality such as \(a - 5 > 6\), the goal is to make \(a\) the subject of the inequality without changing its balance.

• The first step is to "undo" any operations that are being applied to the variable. In the exercise above, the operation applied to \(a\) is subtraction of 5.
• To isolate \(a\), we add 5 to both sides of the inequality. This method of adding or subtracting the same number from both sides is crucial because it maintains the inequality’s validity.

This process transforms the inequality from \(a - 5 > 6\) to \(a > 11\), meaning that we’ve successfully isolated \(a\). This adjustment forms the heart of many algebraic manipulations when dealing with inequalities.

Solving Inequalities

Solving inequalities consists of determining the set of all potential solutions that satisfy the given inequality. The process is quite similar to solving equations, with a few differences due to the nature of inequalities.
When solving an inequality like \(a - 5 > 6\), after isolating the variable, we derive \(a > 11\). This means that any number greater than 11 can be a part of the solution set.

• The sign of inequality (like \(>\), \(<\), \(\geq\), or \(\leq\)) is maintained unless both sides are multiplied or divided by a negative number, which reverses the inequality sign.
• Keep in mind the properties of the inequality when performing operations, such as ensuring consistent operations across both sides of the inequality to maintain balance.

This structure allows us to understand which values fulfill the conditions set by the inequality and represents a vital skill set in different areas of mathematics.

Verification of Solutions

Verification is the process of ensuring that the solution we obtained actually satisfies the original inequality. It acts as a validation step to confirm the solution’s accuracy and reliability.
Let's verify the solution for \(a - 5 > 6\) after determining it as \(a > 11\). We can choose any number greater than 11; for simplicity, let’s use \(a = 12\).

• Substitute \(a = 12\) back into the original inequality to check if the inequality holds true.
• This gives \(12 - 5 > 6\), which simplifies to \(7 > 6\).
• The inequality \(7 > 6\) is true, confirming that our solution correctly fits the initial inequality.

Verification might seem like a small part of the process, but it is essential for ensuring no oversight occurred while isolating the variable or transforming the inequality. This step reinforces the accuracy of mathematical problem solving, making it a cornerstone of reliable solution processes.