Evaluate the inequality $\frac{x}{2} + \frac{3}{4} \leq \frac{5x}{6} - \frac{7}{12}$

Analysis

To solve this linear inequality, we need to isolate $x$. The best approach is to clear the fractions first by finding the Least Common Multiple (LCM) of the denominators (2, 4, 6, and 12).

Concept

When dealing with inequalities involving fractions, multiplying the entire inequality by the LCM of the denominators simplifies the expression into a standard linear inequality. Remember the golden rule: If you multiply or divide an inequality by a negative number, you must reverse the inequality sign.

Step-by-Step Solution

Step 1: Find the LCM of the denominators. The denominators are 2, 4, 6, and 12. The LCM is 12.

Step 2: Multiply every term by 12 to clear the fractions.

Step 3: Simplify each term.

Step 4: Group the like terms. Subtract $6x$ from both sides:

Add 7 to both sides:

Step 5: Solve for $x$. Divide both sides by 4:

This is the same as:

Conclusion

Comparing our result to the given options, we find that our solution matches option A.

Exam Tip

When you have an inequality with many fractions, don't try to find a common denominator for the fractions themselves (which can get messy). Instead, multiply the entire equation by the LCM of all denominators immediately. This turns the problem into a simple whole-number equation, which is much faster and less prone to calculation errors!