### Home > CAAC > Chapter 2 > Lesson 2.1.6 > Problem 2-63

Decide if each of the statements below is sometimes true, always true, or never true.

**Justify**your conclusion. If the statement is not always true, produce a counterexample (an example that shows the statement can or must be false) to help support your claim. Homework Help ✎If

*y*= 3, then 2*y*= 6.If

*x*+ 3 = 9, then*x*= 2.If the product of two numbers is positive (meaning more than zero), then both numbers must be positive.

If

*a*+*b*=*b*, then*a*= 0.If

*a*and*b*are both odd integers, then*ab*is even.If

*x*is greater than zero, then −*x*is less than zero.If

*a*is**prime**(meaning that*a*is an integer greater than 1 and has no other factors besides 1 and itself), then (*a*+ 1) is not prime.If

*x*is greater than*y*, and both*x*and*y*are not zero, thenis greater than .

Substitute the *y*-value into the equation

2(3) = 6

6 = 6

Always true because if *a* = *b*, then *ac* = *bc*.

Substitute the *x* value into the equation.

2 + 3 = 9

Never true, because 2 + 3 ≠ 9 since 5 ≠ 9.

Is there another way to create a positive product, without using positive numbers?

Sometimes true, because a (positive) · (positive) = (positive) but a (negative) · (negative) = (positive) also.

Review the properties of equality studied earlier in this chapter.

Always true, because according to the Identity Property of Addition 0 + *b* = *b* so *a* = 0.

Try a few examples to see if a pattern emerges.

(3)(7) = ?

(5)(11) = ?

(7)(11) = ?

Never true, because the product of two odds is always odd.

Always true, because if *x* is positive −*x* will always be the opposite which is negative.

So the answer will be less than zero.

Try a few examples of prime numbers starting with 2.

2 + 1 = ?

3 + 1 = ?

5 + 1 = ?

7 + 1 = ?

What do you notice?

Sometimes true, because there are instances where the sum is prime(when *a* = 2) and instances where the sum is not prime.

Make sure to try values of *x* and *y* that are positive and negative, where *x* > *y*.

Are these situations true or false?

*x* = 3 and *y* = 2 so, *x* > *y*, then

*x* = −2 and *y* = −5, so *x* > *y*, then

Always true.