Solving Fractional Inequalities Using the Least Common Denominator

Q1. Solve the inequality:

$\frac{5}{6} + \frac{3}{7} \leq \frac{x}{42}$

Background

Topic: Solving Linear Inequalities with Fractions

This question tests your ability to solve inequalities that contain fractions by clearing the fractions using the least common denominator (LCD), then isolating the variable.

Key Terms and Formulas

• Least Common Denominator (LCD): The smallest number that is a common multiple of the denominators in the fractions.

• Distributive Property: $a(b + c) = ab + ac$

• Solving Inequalities: Use algebraic steps similar to solving equations, but remember to reverse the inequality if you multiply or divide by a negative.

Step-by-Step Guidance

1. Identify the denominators in the inequality: $6$, $7$, and $42$.

2. Find the LCD of $6$, $7$, and $42$. The LCD is $42$.

3. Multiply both sides of the inequality by $42$ to clear the fractions:

   $42 \left( \frac{5}{6} + \frac{3}{7} \right) \leq 42 \left( \frac{x}{42} \right)$

4. Apply the distributive property to simplify the left side:

   $42 \cdot \frac{5}{6} + 42 \cdot \frac{3}{7}$

5. Simplify each term and combine like terms to get an expression involving $x$.

Try solving on your own before revealing the answer!