Compound Inequalities - Pre Algebra

1. Solve $-6 < -3x < 9$.

   $-3 < x < 2$. Dividing all three parts by $-3$ reverses both symbols: $2 > x > -3$.

2. What is the solution of $x > 2$ and $x > 5$, written as one inequality?

   $x > 5$. Only numbers greater than 5 satisfy both conditions.

3. Which numbers satisfy $x < 1$ or $x > 4$?

   Every number less than 1 or greater than 4. The numbers from 1 to 4, inclusive, are excluded.

4. Solve $-4 \le 2x \le 10$.

   $-2 \le x \le 5$. Divide all three parts by 2.

5. Solve $1 \le 2x - 3 < 7$.

   $2 \le x < 5$. Add 3 to all parts to get $4 \le 2x < 10$, then divide by 2.

6. What compound inequality describes a temperature $T$ that is at least 10 degrees and less than 25 degrees?

   $10 \le T < 25$.

7. What does the word "and" mean in a compound inequality?

   A value must satisfy both inequalities. The solution is the overlap (intersection) of the two sets.

8. What is the solution of $x < 1$ and $x > 4$?

   There is no solution, because no number is both less than 1 and greater than 4.

9. What is the solution of $x > 2$ or $x > 5$, written as one inequality?

   $x > 2$. The union includes every number in either set, and $x > 5$ lies inside $x > 2$.

10. Write $x > -2$ and $x \le 5$ as a single compound inequality.

    $-2 < x \le 5$.

11. Solve $3 < x + 1 < 7$.

    $2 < x < 6$. Subtract 1 from all three parts.

12. What does the word "or" mean in a compound inequality?

    A value must satisfy at least one of the inequalities. The solution is the union of the two sets.

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