You have correctly divided the compound inequalities into two separate problems and solved them individually.
In the first problem, 12 < 12 + 4x < 0, you found that 4x > 0 and x > 0, which means that x must be greater than 0. In addition, you found that 12 + 4x < 0, which implies 4x < -12, and therefore x < -3. So, combining these two inequalities, you determined that x can be either less than -3 or greater than 0.
In the second problem, 12 - x > 15 or 7x - 13 > 1, you correctly solved them individually by isolating x.
For 12 - x > 15, you subtracted 12 from both sides and obtained -x > 3. By multiplying both sides by -1 and reversing the inequality sign, you correctly found x < -3.
For 7x - 13 > 1, you added 13 to both sides, resulting in 7x > 14. By dividing both sides by 7, you determined that x > 2.
Combining the solutions, you correctly concluded that x can be either less than -3 or greater than 2.
When describing the compound inequalities, you can use the vocabulary words as follows:
- "And" is used when both inequalities must be satisfied. In the first problem, you used "and" to connect the two inequalities 4x > 0 and 12 + 4x < 0, indicating that both conditions need to be true.
- "Or" is used when at least one of the inequalities must be satisfied. In the second problem, you used "or" to separate the two inequalities 12 - x > 15 and 7x - 13 > 1, informing that either one of them being true is sufficient.
- "Intersection" refers to the values that satisfy both inequalities. In the first problem, you found the intersection of the two inequalities to be x < -3.
- "Union" refers to the values that satisfy at least one of the inequalities. In the first problem, you found the union of the two inequalities to be x < -3 or x > 0.
Overall, you have correctly solved the compound inequalities and utilized the vocabulary words in an appropriate manner.