Then a = c. 1 ?. (7+5) + 1 = 7+ ? C. 6 = 6 16 D. 7(x-3) = 7x- ?? One Example (NOT from class!) 4 E. x + (-x) = 0 8 F. if y = 3, then 3 2 2 G. (a + b) + 0 = 9 H. if x = -1, and 3 I. 4x-1=4x 11. Is the set of integers closed under division? Explain why or why not. If it is not closed, give an example. duide 2. Is the set of real numbers closed under addition? Explain why or wh If it is not closed, give an example.
The Correct Answer and Explanation is:
Let’s break down each part of the question:
A. a=ca = ca=c. What is 111?
This seems to be asking for a relationship or equation that involves aaa and ccc, but the statement is incomplete or unclear. Could you provide more context here?
B. (7+5)+1=7+?(7 + 5) + 1 = 7 + ?(7+5)+1=7+?
To solve this equation:
- The left side: (7+5)+1=12+1=13(7 + 5) + 1 = 12 + 1 = 13(7+5)+1=12+1=13.
- The right side: 7+?7 + ?7+?, so we want to find the value of ??? that makes both sides equal.
- Therefore, 7+?=137 + ? = 137+?=13, so ?=6? = 6?=6.
C. 6=66 = 66=6
This is just a statement of equality, which is true. It’s not an equation to solve, just a given fact.
D. 7(x−3)=7x−??7(x – 3) = 7x – ??7(x−3)=7x−??
To expand the left side:
- 7(x−3)=7x−217(x – 3) = 7x – 217(x−3)=7x−21.
So, the equation becomes: - 7x−21=7x−??7x – 21 = 7x – ??7x−21=7x−??, which means ??=21?? = 21??=21.
E. x+(−x)=0x + (-x) = 0x+(−x)=0
This is an identity. Adding a number and its opposite always equals zero. Therefore:
- The equation simplifies to x−x=0x – x = 0x−x=0, which is true for any value of xxx.
F. If y=3y = 3y=3, then 3×2=23 \times 2 = 23×2=2?
This seems to be either a typo or an incomplete question. If you meant to say 3×y=23 \times y = 23×y=2, then:
- When y=3y = 3y=3, 3×3=93 \times 3 = 93×3=9, not 2.
G. (a+b)+0=9(a + b) + 0 = 9(a+b)+0=9
This is another identity. Adding zero to any number leaves the number unchanged:
- So, (a+b)+0=a+b(a + b) + 0 = a + b(a+b)+0=a+b.
To make this equal to 9, a+b=9a + b = 9a+b=9.
H. If x=−1x = -1x=−1, and 333?
This part of the question is unclear. If you meant something like “If x=−1x = -1x=−1, then 3x=?3x = ?3x=?”, we can substitute x=−1x = -1x=−1 into the equation:
- 3x=3(−1)=−33x = 3(-1) = -33x=3(−1)=−3.
I. 4x−1=4x4x – 1 = 4x4x−1=4x
This is an equation we can solve for xxx:
- Subtract 4x4x4x from both sides: 4x−1−4x=4x−4x,4x – 1 – 4x = 4x – 4x,4x−1−4x=4x−4x, which simplifies to: −1=0,-1 = 0,−1=0, which is a contradiction. This means there is no solution.
11. Is the set of integers closed under division? Explain why or why not. If it is not closed, give an example.
The set of integers is not closed under division. This means that when you divide two integers, the result may not be an integer. For example:
- 3÷2=1.53 \div 2 = 1.53÷2=1.5, which is not an integer.
Thus, the set of integers is not closed under division.
2. Is the set of real numbers closed under addition? Explain why or why not.
Yes, the set of real numbers is closed under addition. This means that the sum of any two real numbers is always a real number. For example:
- 3.5+4.2=7.73.5 + 4.2 = 7.73.5+4.2=7.7, which is also a real number.
So, real numbers are closed under addition because the sum of any two real numbers will always yield another real number.
