If r>0, then for all p and q such that pq does not equal 0 and pr>qr we have
a. -p > -q
b. -p > q
c. 1 > -q/p
d. 1 < q/p
e. none of these
This one I do not get at all, either. Please walk me through it. Thank you!
To prove the statement, "If r > 0, then for all p and q such that pq ≠ 0 and pr > qr," we need to go through each option and determine if it is true or false.
a. -p > -q
To check if this statement holds, let's assume -p < -q. Now, multiply both sides of the inequality by r (since r > 0) to maintain the direction:
-rp > -rq
Now, we rearrange the inequality by multiplying both sides by -1 to flip the sign:
rp < rq
By the given condition pr > qr, we know that rp is greater than rq. Hence, our assumption -p < -q is incorrect.
b. -p > q
Similar to the previous case, assume -p < q. If we multiply both sides by r, we'll obtain:
-rp < rq
Flipping the sign:
rp > -rq
Since pr > qr, we can conclude that rp is greater than -rq. This means our assumption -p < q is incorrect.
c. 1 > -q/p
To analyze this equation, we need to consider the case when p is positive and when p is negative.
If p > 0, then q/p < 0. Multiplying both sides by -1 gives us -q/p > 0, which means 1 > -q/p.
If p < 0, then q/p > 0. So, -q/p < 0, which implies 1 is not greater than -q/p.
Since the statement does not hold for all possible values of p, this option is false.
d. 1 < q/p
Similar to the previous case, analyzing both situations when p is positive and when p is negative will provide the answer.
If p > 0, then q/p > 0. Hence, 1 is less than q/p.
If p < 0, then q/p < 0. So, 1 is not less than q/p.
Since the statement does not hold for all possible values of p, this option is false.
After evaluating each option, we find that none of the given options (a, b, c, d) are true. Therefore, the correct answer is e. "None of these."