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Which of the following equations has the set of all real numbers as its solution set?
- a)3(N+4)+5N=8(N+3)
- b)4(N+4)+4N=8(N+3)
- c)2(N+4)+6N=8(N+3)
- d)6(N+4)+2N=8(N+3)
- e)5(N+4)+3N=8(N+3)
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Which of the following equations has the set of all real numbers as it...
The right side of each equation is 8(N+3), which simplifies by way of distribution to
8(N+3)=8⋅N+8⋅3=8N+24
If the left side of the equation simplifies to an identical expression, the equation has all real numbers as its solutions.
We test the left side of each equation:
2(N+4)+6N=8(N+3)
2(N+4)+6N=2⋅N+2⋅4+6N=2N+8+6N=8N+8
2(N+4)+6N=2⋅N+2⋅4+6N=2N+8+6N=8N+8
3(N+4)+5N=8(N+3)
3(N+4)+5N=3⋅N+3⋅4+5N=3N+12+5N=8N+12
4(N+4)+4N=8(N+3)
4(N+4)+4N=4⋅N+4⋅4+4N=4N+16+4N=8N+16
5(N+4)+3N=8(N+3)
5(N+4)+3N=5⋅N+5⋅4+3N=5N+20+3N=8N+20
6(N+4)+2N=8(N+3)
6(N+4)+2N=6⋅N+6⋅4+2N=6N+24+2N=8N+24
Of the given choices,
6(N+4)+2N=8(N+3)
can be rewritten as
8N+24=8N+24,
which is an identity and has the set of all real numbers as its solution set.
Most Upvoted Answer
Which of the following equations has the set of all real numbers as it...
Explanation:
Identifying the correct equation:
To find the equation with the set of all real numbers as its solution set, we need to simplify the equations and check for any restrictions on the variable N.
Solving equation (d):
6(N+4) + 2N = 8(N+3)
6N + 24 + 2N = 8N + 24
8N + 24 = 8N + 24
Since the equation simplifies to a true statement, it has all real numbers as its solution set.
Checking other options:
a) 3(N+4) + 5N = 8(N+3)
Simplifying: 3N + 12 + 5N = 8N + 24
This equation does not hold true for all real numbers.
b) 4(N+4) + 4N = 8(N+3)
Simplifying: 4N + 16 + 4N = 8N + 24
This equation also does not hold true for all real numbers.
c) 2(N+4) + 6N = 8(N+3)
Simplifying: 2N + 8 + 6N = 8N + 24
This equation does not hold true for all real numbers.
e) 5(N+4) + 3N = 8(N+3)
Simplifying: 5N + 20 + 3N = 8N + 24
This equation also does not hold true for all real numbers.
Therefore, the correct equation with the set of all real numbers as its solution set is option 'd'.
Identifying the correct equation:
To find the equation with the set of all real numbers as its solution set, we need to simplify the equations and check for any restrictions on the variable N.
Solving equation (d):
6(N+4) + 2N = 8(N+3)
6N + 24 + 2N = 8N + 24
8N + 24 = 8N + 24
Since the equation simplifies to a true statement, it has all real numbers as its solution set.
Checking other options:
a) 3(N+4) + 5N = 8(N+3)
Simplifying: 3N + 12 + 5N = 8N + 24
This equation does not hold true for all real numbers.
b) 4(N+4) + 4N = 8(N+3)
Simplifying: 4N + 16 + 4N = 8N + 24
This equation also does not hold true for all real numbers.
c) 2(N+4) + 6N = 8(N+3)
Simplifying: 2N + 8 + 6N = 8N + 24
This equation does not hold true for all real numbers.
e) 5(N+4) + 3N = 8(N+3)
Simplifying: 5N + 20 + 3N = 8N + 24
This equation also does not hold true for all real numbers.
Therefore, the correct equation with the set of all real numbers as its solution set is option 'd'.
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