Solving Compound Inequalities with 'Or' (Union of Solution Sets)

Q1. Solve the compound inequality: 6(x - 5) < 12 or x + 7 > 10

Background

Topic: Compound Inequalities (Union)

This question tests your understanding of how to solve compound inequalities connected by the word "or." You need to solve each inequality separately and then find the union of their solution sets.

Key Terms and Formulas

• Compound Inequality ("or"): A value is a solution if it satisfies at least one of the inequalities.

• Union (U): The set of all values that are solutions to either inequality.

• Interval Notation: Used to express the solution set on the number line.

Step-by-Step Guidance

1. Solve the first inequality: $6(x - 5) < 12$

   • Divide both sides by 6 to isolate $(x - 5)$:

   • $\frac{1}{6} \cdot 6(x - 5) < \frac{1}{6} \cdot 12$

   • $x - 5 < 2$

2. Add 5 to both sides to solve for $x$:

   • $x - 5 + 5 < 2 + 5$

   • $x < 7$

3. Solve the second inequality: $x + 7 > 10$

   • Subtract 7 from both sides:

   • $x + 7 - 7 > 10 - 7$

   • $x > 3$

4. Express each solution in interval notation:

   • For $x < 7$, the interval is $(-\infty, 7)$

   • For $x > 3$, the interval is $(3, \infty)$

5. To find the solution to the compound inequality, take the union of the two intervals: $(-\infty, 7) \cup (3, \infty)$

Try solving on your own before revealing the answer!

Final Answer: $(-\infty, \infty)$

The union of $(-\infty, 7)$ and $(3, \infty)$ covers all real numbers, so the solution set is $(-\infty, \infty)$.

This means every real number is a solution to at least one of the inequalities.