Solving Compound Inequalities and Expressing Solutions in Interval Notation

Q1. Solve the compound inequality and write the solution in interval notation:

$x + 11 \geq 3$ and $7x - 12 \geq 2$

Background

Topic: Compound Inequalities and Interval Notation

This question tests your ability to solve compound inequalities, interpret the word "and" as the intersection of solution sets, and express the final solution in interval notation. You will need to solve each inequality separately, then find the intersection of their solution sets.

Key Terms and Formulas

• Compound Inequality ("and"): The solution is the set of values that satisfy both inequalities at the same time (intersection).

• Interval Notation: A way to represent all numbers between two endpoints. For example, $[a, b)$ means all $x$ such that $a \leq x < b$.

Step-by-Step Guidance

1. Solve the first inequality: $x + 11 \geq 3$ Subtract 11 from both sides to isolate $x$.

2. Solve the second inequality: $7x - 12 \geq 2$ Add 12 to both sides, then divide both sides by 7 to solve for $x$.

3. Write each solution in interval notation: Express the solution to each inequality as an interval.

4. Find the intersection: The solution to the compound inequality is the set of values that are in both intervals. Draw a number line or compare the intervals to find the overlap.

Try solving on your own before revealing the answer!

Final Answer: $[2, \infty)$

The intersection of the two solution sets is $[2, \infty)$, since $x \geq -8$ and $x \geq 2$ overlap at $x \geq 2$.

In interval notation, this is written as $[2, \infty)$.