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The addition principle is crucial when solving inequalities. It means you can add the same number to both sides of an inequality without changing its truth. Let's dive in.

In the exercise, we start with the inequality: \[ x + \frac{1}{4} \leq \frac{1}{2} \]

This principle helps in moving terms around to isolate the variable. By adding or subtracting the same value on each side, we maintain the inequality's balance. In this specific problem, you subtract \(\frac{1}{4}\) from both sides to keep the inequality true while simplifying it. Compare it to balancing a scale; if you take the same weight off both sides, it stays balanced.

Isolating the variable is a critical goal when solving inequalities. This process involves manipulating the expression so that the variable is by itself on one side of the inequality.

In our exercise, isolating the variable means we need to get 'x' alone on one side of the inequality:

Start with:\[ x + \frac{1}{4} \leq \frac{1}{2} \]

Subtract \( \frac{1}{4} \) from both sides:

\[ x + \frac{1}{4} - \frac{1}{4} \leq \frac{1}{2} - \frac{1}{4} \]

This leads us to:

\[ x \leq \frac{1}{4} \]

Through this process, 'x' is isolated, allowing us to easily understand the solution.

Isolating the variable enables you to determine the range or specific value the variable can take to make the inequality true.

Combining like terms simplifies mathematical expressions, making them easier to work with.

In our example, look at the right-hand side of the inequality after subtraction:
\[ x \leq \frac{1}{2} - \frac{1}{4} \]

To combine like terms, remember that both fractions should have a common denominator. Here, the common denominator is 4:
\[ \frac{1}{2} = \frac{2}{4} \]

Replacing \( \frac{1}{2} \) with \( \frac{2}{4} \):
\[ x \leq \frac{2}{4} - \frac{1}{4} \]

Perform the subtraction:
\[ x \leq \frac{1}{4} \]

Combining like terms makes the inequality clear and more straightforward to solve. It brings the result from being an abstract idea to a clear, solvable form.