Question

Make and test a conjecture about the given quantity.

18. The product of two negative even integers
19. The sum of two negative odd integers
20. The sum of two even integers and two odd integers
21. When you return from a run, your
friend states, “If you ran 1 mile, then
you ran 5280 feet." Your friend then
asks if you can write the statement as
a biconditional statement. Write the
biconditional statement and decide
whether it is true.

Answer 1

18. The conjecture is: The product of two negative even integers is an even integer.

Testing: Let integers be -4 and -6. [Two negative even integers]

Then (-4) × (-6) = 24 [Even integer]

19. The conjecture is: Sum of two negative odd integers is a negative even integer.

Testing: Let the two negative odd integers be -3 and -7.

Then -3 + (-7) = -10 is a negative even integer

20. The conjecture is: Sum of two even integers and two odd integers is an even integer.

Testing: Let integers be 4, 12, 13, 17 (Two even integers and two odd integers)

Then 4 + 12 + 13 + 17 = 46 (Even integer)

21. The statement is: p------->q [If you ran 1 mile, then you ran 5280 feet]

Decide whether statement is true: So if p is True then q is also True.

Hence p------->q is True. So the given statement is True.

Explanation:

18. The product of two negative even integers.

Let the integers be -2m and -2n, m and n are positive integers

Product in (-2m) × (-2n) = 4mn = 2(2mn)

= 2p, p = 2mn

The conjecture is: The product of two negative even integers is an even integer.

Testing: Let integers be -4 and -6. [Two negative even integers]

Then (-4) × (-6) = 24 [Even integer]

19. The sum of two negative odd integers.

Let the integers be -(2m + 1) and -(2n + 1), m and n are positive integers

Then -(2m + 1) + [-(2n + 1)] = -2m - 1 - 2n - 1

= -2m - 2n - 2

= -2(+m + 2 + 1)

= -2p, p = m + n + 1

The conjecture is: Sum of two negative odd integers is a negative even integer.

Testing: Let the two negative odd integers be -3 and -7.

Then -3 + (-7) = -10 is a negative even integer

20. The sum of two even integers and two odd integers.

Let the integers be 2m, 2n, 2p + 1, 2q + 1 (m, n, p, q are positive integers)

Then, 2m + 2n + 2p + 1 + 2q + 1 = 2(m + n + p + q + 1)

= 2r, r = m + n + p + q + 1

The conjecture is: Sum of two even integers and two odd integers is an even integer.

Testing: Let integers be 4, 12, 13, 17 (Two even integers and two odd integers)

Then 4 + 12 + 13 + 17 = 46 (Even integer)

21. Let p: You ran 1 mile

Let q: You 5280 feet

The statement is: p------->q [If you ran 1 mile, then you ran 5280 feet]

Since 1 mile = 5280 feet

So if p is True then q is also True.

Hence p------->q is True. So the given statement is True.