Compound Inequalities




Definition Words

◦Compound inequalities

◦And

◦Or

◦Intersection

◦Union



1). 12 < 12 + 4x < 0

2). 12 – x > 15 or 7x – 13 > 1

I divide it up the compound inequalities into two problems

12 <12+ 4x

4 x >0

or

x >0

and

12+4x <0

4 x < -12

x < - 3

so x is either left of -3 or right of 0 on a number line

12 - x > 15

-x > 3

I then multiplied both sides by -1 and changed the direction of the arrow when I changed the signs.

x < - 3



7 x - 13 > 1

7 x > 14

x > 2



so x is left of -3 or right of +2

I need someone to check and see if this is correct and how do I use the vocabulary words correctly. I'm having a difficult time with this.

There is no solution to #1 since 12 is not less than 0.

As written
12 < 12 + 4x < 0
means 12 < 12+4x AND 12+4x < 0
To find such a number, you would need
12 < 0

12 – x > 15 or 7x – 13 > 1

12-x > 15
-3 > x

7x-13 > 1
7x > 14
x > 2

You are correct

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.