I did these, but I got the wrong answers.
Please explain how ...
1. If xy>1 and z<0, which of the following statements must be true?
I. x>z
II.xyz<-1
III.xy/z<1/z
A. I. only
B. II. only
C. III. only
D. II. and III.
E. None
2.
ab>0
1/a<1/b
Which is greater, a or b?
3.
p/4<q/12
Which is greater, (p-1)/4 or (q-9)/12?
4.
a and b are positive integers.
7<ab<21
Which is greater, a+b or ab?
5.
-1<=x-3<=6
5<=y+1<=9
Which is greater, the largest possible value of x/y or 2?
6.
a+b=5
b<7/3
2. a is greater because it has the smaller reciprocal.
3. P/4<q/12
Multiply both sides by 12:
3P<q or q>3P.
(P-1)/4
Let P = 3
(3-1)/4 = 2/4 = 1/2.
(q-9)/12
Let q = 10 which greater than 3P.
(10-9)/12 = 1/12.
Therefore, (P-1)/4>(q-9)/12. This is
true for values of q up to 14.
4. 7<ab<21
This a compound inequality which states
that ab is greater than 7 but less than
21. Therefore, ab can be any whole number ranging from 8 to 20.
When ab = 8 = 1*8 = 2*4.
a+b = 1+8 = 9.
a+b = 2+4 = 6.
When ab = 20 = 1*20 = 2*10 = 4*5
a+b = 1+20 = 21
a+b = 2+10 = 12
The results show that a+b><ab depending
on the value of a and b.
5. -1<=x-3<=6. 5<=y+1<=9
2<=X<=9
Xmin = 2, Xmax = 9.
5<=y+1<=9.
4<=Y<=8
Ymin = 4, Ymax = 8.
X/Y = 9/4 =2 1/4,max.
Therefore,(x/y)>2.
1. To solve this, we need to consider the given information: xy > 1 and z < 0.
I. x > z: Since z is negative, we can multiply both sides of the inequality by z, but then we must reverse the inequality sign. This gives us xz < z^2. Since xy > 1, we can also say that xy > z^2. Combining these inequalities, we have xz < xy, which implies x > z. Therefore, statement I is true.
II. xyz < -1: We know that xy > 1 and z < 0. Since the product of two positive values is positive, and the product of a positive and a negative value is negative, we can conclude that xyz is negative. Therefore, statement II is true.
III. xy/z < 1/z: We know that z < 0, so 1/z is also negative. Dividing both sides of the given inequality by z results in xy < 1, which means xy is less than 1. Thus, statement III is true.
Therefore, the correct answer is (D) II. and III.
2. To determine whether a or b is greater, we need to consider the given information: ab > 0 and 1/a < 1/b.
If ab > 0, it means that a and b have the same sign (both positive or both negative), since the product of a positive and a negative number is negative.
Since 1/a < 1/b, we can cross-multiply to get b < a.
Combining the information that a and b have the same sign and b < a, we can conclude that a > b.
Therefore, a is greater than b.
3. To determine which is greater between (p-1)/4 and (q-9)/12, we need to compare the given inequality p/4 < q/12.
First, let's cross-multiply to eliminate the fractions:
12p < 4q
To isolate p, divide both sides by 12:
p < (4/12)q
Reduce the fraction to get:
p < (1/3)q
Now, let's consider (p-1)/4.
Since p < (1/3)q, p-1 < (1/3)q - 1/3.
Multiplying both sides by 4 gives us:
4p - 4 < (4/3)q - (4/3)
Simplifying further, we have:
4p - 4 < (4/3)q - (4/3)
Now, divide both sides by 4 to isolate p:
p - 1 < (1/3)q - (1/3)
Add 1 to both sides:
p < (1/3)q
Comparing the inequalities (p < (1/3)q) and ((p-1)/4 < (q-9)/12), we can see that they are equivalent. Therefore, neither expression is greater than the other.
4. We are given the inequality 7 < ab < 21, and we need to determine whether a+b or ab is greater.
Since a and b are positive integers, we know that both a and b are greater than 1.
To maximize the value of a+b, we can assign the smallest possible value to a (which is 2) and the smallest possible value to b (which is 3). In this case, a+b = 2+3 = 5.
To maximize the value of ab, we can assign the smallest possible value to a (which is 2) and the largest possible value to b (which is 6). In this case, ab = 2*6 = 12.
Therefore, ab is greater than a+b.
5. The given inequalities are -1 ≤ x-3 ≤ 6 and 5 ≤ y+1 ≤ 9. We need to determine whether the largest possible value of x/y or 2 is greater.
First, let's solve the inequalities individually:
-1 ≤ x-3 ≤ 6 becomes -1+3 ≤ x ≤ 6+3, which simplifies to 2 ≤ x ≤ 9.
Similarly, 5 ≤ y+1 ≤ 9 becomes 5-1 ≤ y ≤ 9-1, which simplifies to 4 ≤ y ≤ 8.
Now, let's find the largest possible value of x/y:
To maximize x/y, we need to maximize x and minimize y.
The largest value of x is 9 (from the inequality 2 ≤ x ≤ 9), and the smallest value of y is 4 (from the inequality 4 ≤ y ≤ 8).
Therefore, the largest possible value of x/y is 9/4.
Comparing 9/4 to 2, we can see that 9/4 is greater than 2.
Therefore, the largest possible value of x/y is greater than 2.
6. We are given the equations a+b = 5 and b < 7/3. We need to determine whether a or b is greater.
From the first equation, we can solve for a by subtracting b from both sides:
a = 5 - b
Now, let's substitute this into the second equation:
b < 7/3
5 - b < 7/3
To solve for b, we first need to eliminate the fraction. Multiply both sides by 3 to get:
15 - 3b < 7
Next, subtract 15 from both sides:
-3b < -8
Divide both sides by -3, but remember to reverse the inequality sign since we're dividing by a negative number:
b > 8/3
Therefore, b is greater than a.