91Ó°ÊÓ

Algebraic manipulation is a fundamental skill in solving inequalities. It involves rearranging and simplifying algebraic expressions to make them easier to work with. In our example, the inequality is: \(\frac{f}{2} + 5 > 10\). We need to make the left side simpler. This involves a few key operations like addition, subtraction, multiplication, and division. These operations must be applied correctly to maintain the balance of the inequality. For instance, in our problem, the first step is to subtract 5 from both sides. This ensures that the inequality remains balanced: \(\frac{f}{2} + 5 - 5 > 10 - 5\) Simplifying this, we get: \(\frac{f}{2} > 5\). Remember, the goal is to simplify the expression so that you can isolate the variable.

Isolating variables is another crucial step when solving inequalities. The aim is to 'isolate' the variable on one side of the equation. This makes it easier to solve for the variable. In our example: \(\frac{f}{2} > 5\), we need to isolate \(f\). To do this, we get rid of the coefficient (in this case, \(1/2\)) attached to \(f\). We achieve this by performing the inverse operation. Here, we multiply both sides of the inequality by 2: \(\frac{f}{2} \times 2 > 5 \times 2\). Simplifying this, we get: \(f > 10\). Now the variable \(f\) is isolated, which makes the solution straightforward: any value of \(f\) that is greater than 10 satisfies the inequality.

The steps to solve an inequality follow a logical sequence. Here's a breakdown:

• Identify the inequality: In this problem, it's \(\frac{f}{2} + 5 > 10\)
• Isolate the term with the variable: Subtract 5 from both sides: \(\frac{f}{2} > 5\)
• Remove coefficients: Multiply both sides by 2 to isolate \(f\): \(f > 10\)
• Verify the solution: Check that the variable satisfies the inequality.

Each step involves careful algebraic manipulation to maintain balance. Always remember:
- Perform operations on both sides of the inequality.
- Think about inverse operations for isolating the variable.
- Verify your solution to ensure it's correct. By following these steps, solving inequalities becomes systematic and straightforward.