Solving a Linear Inequality with Fractions and Writing the Solution in Interval Notation

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

\[ \frac{7x + 1}{3} - \frac{4x - 5}{4} \leq -2 \]

Background

Topic: Linear Inequalities with Fractions

This question tests your ability to solve linear inequalities that involve fractions. You will need to clear the fractions, combine like terms, and use properties of inequalities to isolate the variable. Finally, you will express the solution in interval notation.

Key Terms and Formulas

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

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

• Addition Property of Inequality: If \( a < b \), then \( a + c < b + c \).

• Multiplication Property of Inequality: If \( a < b \) and \( c > 0 \), then \( ac < bc \). If \( c < 0 \), the inequality sign reverses.

• Interval Notation: A way to describe the set of solutions using parentheses and brackets.

Step-by-Step Guidance

1. Clear the fractions: Identify the least common denominator (LCD) of 3 and 4, which is 12. Multiply both sides of the inequality by 12 to eliminate the denominators.

   \[ 12 \left( \frac{7x + 1}{3} - \frac{4x - 5}{4} \right) \leq 12(-2) \]

2. Distribute and simplify: Apply the distributive property to remove parentheses and simplify each term.

   \[ 4(7x + 1) - 3(4x - 5) \leq -24 \]

3. Combine like terms: Expand and combine the terms involving \( x \) and the constants.

   \[ 28x + 4 - 12x + 15 \leq -24 \]

   Combine the \( x \) terms and constants.

4. Isolate the variable term: Use the addition property of inequality to move constants to one side and terms with \( x \) to the other.

   Subtract 19 from both sides to further isolate the variable term.

   \[ 16x + 19 \leq -24 \]

   \[ 16x \leq -43 \]

5. Solve for \( x \): Divide both sides by 16 (a positive number, so the inequality direction stays the same) to isolate \( x \).

   \[ x \leq \frac{-43}{16} \]

Try solving on your own before revealing the answer!

Final Answer:

\[ (-\infty, \frac{-43}{16}] \]

This interval notation shows all real numbers less than or equal to \( \frac{-43}{16} \).