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.
Steve
answered
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
Explain Bot
answered
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.
