Solving Linear Inequalities with Fractions and Interval Notation

Q1. Solve the following inequality and write the solution set using interval notation:

$\frac{x+3}{16} + \frac{x-5}{10} < \frac{1}{2}$

Background

Topic: Linear Inequalities with Fractions

This question tests your ability to solve linear inequalities that involve fractional terms, clear denominators using the least common denominator (LCD), and express the solution in interval notation.

Key Terms and Formulas

• Linear Inequality: An inequality involving a linear expression.

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

• Interval Notation: A way to represent the set of solutions to an inequality.

Step-by-Step Guidance

1. Identify the denominators in the inequality: 16, 10, and 2. Find the LCD, which is 80.

2. Multiply both sides of the inequality by 80 to clear the fractions:

   $80 \left( \frac{x+3}{16} + \frac{x-5}{10} \right) < 80 \left( \frac{1}{2} \right)$

3. Distribute 80 to each term and simplify:

   $5(x+3) + 8(x-5) < 40$

4. Expand and combine like terms:

   $5x + 15 + 8x - 40 < 40$

   $13x - 25 < 40$

Try solving on your own before revealing the answer!

Final Answer: (-∞, 5)

$13x - 25 < 40 \implies 13x < 65 \implies x < 5$

The solution set in interval notation is $(-\infty, 5)$, which means all values less than 5 satisfy the original inequality.