For three consecutive integers the sum of twice the smallest integer and three times

Martin V

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These questions have driven me absolutely mad, as even with the key, I have been unable to understand the thought process behind the answers.

1.) For three consecutive integers the sum of twice the smallest integer and three times the largest integers is 126. What are the three consecutive integers?
Key answer: 24, 25, 26

2.) A student scored 93 on the first test, 85 ont he second test, and 77 on each of the other tests. The student's average test score for the course is 81.8. If each test is equally weighed, what is the total number of tests taken?
Key answer: 5 tests were taken in total.

3.) Given a and b are nonzero real numbers and |a| = -a and |b| = b, which must be positive?
a.) a-b
b.) a+b
c.) b-a
d.) ab
key answer: b

Please help me understand how these answers have been attained! I'm downright clueless here:(!!
 
These questions have driven me absolutely mad, as even with the key, I have been unable to understand the thought process behind the answers.

1.) For three consecutive integers the sum of twice the smallest integer and three times the largest integers is 126. What are the three consecutive integers?
Key answer: 24, 25, 26

My answer: 2x + (x+1) + 3(x+2) = 126
6x + 7 = 126
6x = 119
x = 19.8333...

2.) A student scored 93 on the first test, 85 on the second test, and 77 on each of the other tests. The student's average test score for the course is 81.8. If each test is equally weighed, what is the total number of tests taken?
Key answer: 5 tests were taken in total.

My answer: (93 + 85 + 77x/(x-2)) = 81.8
178 + 77x = 81.8x - 136
341.6 = 4.8x
85.4 = x
85.4 + 2 = 87.2 tests taken in total?

3.) Given a and b are nonzero real numbers and |a| = -a and |b| = b, which must be positive?
a.) a-b
b.) a+b
c.) b-a
d.) ab
key answer: b

My answer: This question has no real solution, as any absolute value can not produce a negative value.

Please help me understand how these answers have been attained! I'm downright clueless here:(!!
 
1.) For three consecutive integers the sum of twice the smallest integer and three times the largest integers is 126. What are the three consecutive integers?

My answer: 2x + (x+1) + 3(x+2) = 126
For what do "x", "x + 1", and "x + 2" stand? How do you get that "the sum of (twice the smallest) and (three times the largest)" is equal to "the sum of (twice the smallest), the (middle), and (three times the largest)"?

2.) A student scored 93 on the first test, 85 on the second test, and 77 on each of the other tests. The student's average test score for the course is 81.8. If each test is equally weighed, what is the total number of tests taken?

My answer: (93 + 85 + 77x/(x-2)) = 81.8
For what does "x" stand? What is your reasoning for the "x/(x - 2)" expression?

3.) Given a and b are nonzero real numbers and |a| = -a and |b| = b, which must be positive?
a.) a-b
b.) a+b
c.) b-a
d.) ab

My answer: This question has no real solution, as any absolute value can not produce a negative value.
Where do you see an absolute value returning a negative value? I see where an absolute value is returning something with an opposite sign, but that's only normal (and required) when the argument itself was negative... ;)
 
3.) Given a and b are nonzero real numbers and |a| = -a and |b| = b, which must be positive?
a.) a-b
b.) a+b
c.) b-a
d.) ab
key answer: b
Your answer key is incorrect. To see that let \(\displaystyle a=-3~\&~b=1\)

There is a correct answer listed. What is it? WHY?
 
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