If 4x+2<x2+3x−18<0, where x is an integer, what is the value of x ?
Given Info:
To Find:
Approach:
Working out:
1st Inequality
⇒x^{2}+6x−3x−18<0
⇒x(x + 6) - 3(x + 6) < 0
⇒ (x + 6)(x - 3) < 0
2nd Inequality
⇒x^{2}+3x−18−4x−2>4x+2−4x−2
⇒x^{2}−x−20>0
⇒x^{2}−5x+4x−20>0
⇒(x−5)(x+4)>0
Correct Answer : Option B
Is the distance between Snape’s home and college greater than 20 kilometers?
(1) If Snape drives at a speed of 25 kilometers per hour, he takes less than 1 hour to reach his college from his home.
(2) If Snape drives at a speed of 12 kilometers per hour, he takes more than 105 minutes to reach his college from his home
Steps 1 & 2: Understand Question and Draw Inferences
Let the distance between Snape’s home and college be d.
We need to find , if d > 20?
Step 3: Analyze Statement 1 independently
(1) If Snape drives at a speed of 25 kilometers per hour, he takes less than 1 hour to reach his college from his home.
d=s∗t
t=d/s Now we know that the time taken by Snape to reach his college from his home is less than 1 hour. So, t < 1.
As t=d/s , we can write d/s<1
d/25<1
d<25
Insufficient to tell us if d > 20 or not.
Step 4: Analyze Statement 2 independently
(2) If Snape drives at a speed of 12 kilometers per hour, he takes more than 105 minutes to reach his college from his home
d=s∗t
The time taken, t is greater than 105 minutes, which can be written as ((1+45) /60) hours
So, t > ((1+45) /60)
t >((1+3)/4)
t > 7/4
t=d/s, we have d/12>7/4
d > 21
Sufficient to tell us that d > 20.
Step 5: Analyze Both Statements Together (if needed)
This step is not needed as we have a unique answer from step -4
Answer: B
For how many values of integer n is 0.01 < 3^{-n+1} < 0.1?
Given:
To find: Number of integral solutions for the inequality 0.01 < 3^{-n+1} < 0.1
Approach:
Working Out:
Equating 3^{n-1} to its possible values (3^{3} and 3^{4})
Looking at the answer choices, we see that the correct answer is Option C
Is x + y > 0?
(1) xy > 0
(2) x^{3}y^{2}>0
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Is, x+ y > 0?
Step 3: Analyze Statement 1 independently
(1) xy > 0
Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) x^{3}y^{2}>0
Insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
Combining both the statements, we have x, y > 0. So, x + y > 0.
Sufficient to answer
Answer: C
Two shopkeepers, Bruce and Wayne, sold the same toy at different prices. If the ratio of the prices at which Bruce and Wayne purchased the toy was 3:4 respectively, who made a greater profit on selling the toy?
(1) The profit percentage of Bruce was higher than that of Wayne.
(2) The ratio of the prices at which Bruce and Wayne sold the toy was 4:3 respectively.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Which is greater out of P_{1 }and P_{2}
Step 3: Analyze Statement 1 independently
(1) The profit percentage of Bruce was higher than that of Wayne.
(For example, if P_{1} = 5 and P_{2} = 4, then P_{1} > P_{2} but if P_{1} = 3.5 and P_{2} = 4, then P_{1} < P_{2}. Both these sets of values for P_{1} and P_{2} satisfy Statement 1)
Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The ratio of the prices at which Bruce and Wayne sold the toy was 4:3 respectively.
Assuming, P_{1 }> P_{2}, we have
Alternate Method
As we can see the selling price of Bruce > selling price of Wayne (4y > 3y) and the cost price of Bruce < cost price of Wayne (3x < 4x).
Hence, the profit made by Bruce > Profit made by Wayne.
Thus P_{1} > P_{2}
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step-4, this step is not required.
Answer: B
How many integers satisfy the inequality x^{2 }-8 >0, where |x| ≤ 5?
Given
To Find: Number of integers that satisfy x^{2} - 8 > 0 ?
Approach
Working Out
1. |x| ≤ 5
2. x^{2} - 8 > 0
3. Hence, the expression is positive for the region x > 2√2 and x < -2√2
4. The integers that satisfy the above and lie between -5 and 5 are {-5, -4, -3, 3, 4, 5}, i.e. a total of 6 integers.
If A and B are non-zero numbers such that AB >0, is |A - B| > |A| - |B|?
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
Thus, from the information given in Statement 1, the answer could be YES or NO.
Since we have not been able to determine a unique answer, Statement 1 is not sufficient.
Step 4: Analyze Statement 2 independently
(2) 2A + B < 0
.
Step 5: Analyze Both Statements Together (if needed)
Since we could arrive at a unique answer, the two statements together are sufficient.
Answer: Option C
Which of the following inequalities have their solutions represented in the shaded part of the number line above?
Given
To Find: The options that have their solution range represented in the number line graph
Approach
Working Out
The graph in the question statement does not capture the complete region of |x| < 5.
2. |x + 3 | ≤ 2
The graph in the question statement does not capture the complete region of |x + 3 | ≤ 2
3.
The graph in the question statement does not capture the complete region of
As the number line graph in the question statement does not capture the complete solution region of any of the expressions in the options, answer is option E
Answer: E
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find: Is a/pq>0 ?
Step 3: Analyze Statement 1 independently
p < q
So, Statement 1 alone is not sufficient.
Step 4: Analyze Statement 2 independently
Since we get the same answer – YES – in each of the two possible cases, Statement 2 is sufficient to arrive at a unique answer.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
If y > x, is y > 64
(1) x has 7 factors(all even but one) and is divisible by only one prime number.
(2) |y-64| > |x-64|
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Is y > 64?
Step 3: Analyze Statement 1 independently
(1) x has 7 factors out of which 6 are even factors and one of the odd factors is 1.x is divisible by only one prime number.
Sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) |y-64| > |x-64|
In this case y > 64
Thus, for all the possible cases, y > 64. Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
A group of 10 athletes was competing in a marathon race. If the range of the distance run by the athletes was 10 kilometers, did the total distance run by the athletes exceed 200 kilometers?
(1) The distance covered by one of the athletes was more than 30 kilometers.
(2) The total distance covered by 5 athletes was 125 kilometers.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: If d_{1} + d_{2} + d_{3}…+ d_{10} > 200?
Step 3: Analyze Statement 1 independently
(1) The distance covered by one of the athletes was more than 30 kilometers.
Sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The total distance covered by 5 athletes was 125 kilometers.
Insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step-3, this step is not required.
Answer: A
Is x < y?
Steps 1 & 2: Understand Question and Draw Inferences
We need to find if x < y
Step 3: Analyze Statement 1 independently
Statement 1 says that
If 1/x and 1/y have the same sign, then the direction of inequality is reversed for their reciprocals, that is, x > y. And so, the answer to the asked question is NO.
If 1/x and 1/y have the opposite sign, then x < y. The answer is YES.
So, not sufficient to determine a unique answer to the posed question.
Step 4: Analyze Statement 2 independently
Statement 2 says that x + y = 10
This can happen for x = 2 and y = 8 (Here, x < y)
Or for x = 11 and y = -1 (Here, x > y)
So, not sufficient to determine if x < y.
Step 5: Analyze Both Statements Together (if needed)
From St. 1,
The answer to the question depends on whether x and y have the same sign or opposite sign
From the analysis done in St. 2:
x and y can either have the same sign or opposite signs.
So, still not sufficient.
Answer: E
If (x−2)^{2}>1,what is the range of values for x?
Given Info:
To Find:
Approach:
Working out:
⇒ (X - 2)^{2} - 1> 0
Answer
Hence answer option D is correct.
If (|x| -1)^{2} > 4, which of the following may be true?
I. |x| < 1
II. |x +1| < 1
III. |x -2|^{2} ≥ 4
Given:
To Find: The options that may be true for the given inequality
Approach
Working Out
2. Solving the reduced inequality
So, the inequality is true for x > 3 and x < -3
I. |x| < 1
II. |x +1| < 1.
III. |x -2|^{2} ≥ 4.
5. |x-2| ≤ -2.
So, the inequality is true for x ≤ 0 and x ≥ 4. This range has some overlap with S.
Hence option III may be true for some values of x.
Correct Answer: Option C
If x2≤16andx2>4, how many integral values of x are possible?
Given
To Find: Integral values of x that satisfy both the inequalities
Approach
into (x+a) (x-a) ≤ 0 form and the inequality x^{2}>4
Working Out
d. So, the inequality is true for range -4 ≤ x ≤ 4
e. Hence, the integral values of x in the range = {-4, -3, -2, -1, 0, 1, 2, 3, 4}
d. So, the inequality is true for range when x < -2 or x > 2
e. Hence, the integral values of x in the range = {-∞…….-4, -3, 3, 4, ………+∞}
3. The integral values of x that satisfy both the inequality are = { -4, -3, 3, 4}, i.e. 4 values.
Answer: D
If a, b, c and d are positive consecutive multiples, not necessarily in that order, of a positive integer x greater than 1, is a + b + c + d ≥ 50?
(1) c = 15
(2) The difference between d and b is divisible by only four positive integers, one of which is 10.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Is a + b + c + d ≥ 50?
That is, is xy + x(y+1) +x(y+2) +x(y+3) ≥ 50 ?
Is 4x +6xy ≥ 50 ?
That is, is x(2y + 3) ≥ 25?
So, we need to find a unique answer to the question is x(2y + 3) ≥ 25?
Step 3: Analyze Statement 1 independently
(1) c = 15
As c is a multiple of x, the value of x can be the factors of 15 greater than 1= {3, 5, or 15}. We need to see, if for a value of x, is x(2y+3) ≥ 25?
As we do not have a unique answer to the question Is, a +b+c+d ≥ 50 , the statement is insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The difference between d and b is divisible by only four positive integers, one of which is 10.
The possible values of difference between d and b can be ={x, 2x, or 3x}. As we are given that the difference between d and b is divisible by 10, we will try to find , if for all values of x and y, is x(2y+3) ≥ 25?
Also, as 10 has four factors (1,2,5 and 10), a number, which is divisible by 10 will be divisible by all the factors of 10. Since |d-b| is divisible by 10 and has only 4 factors, the only possible value of |d-b| = 10
As we have a unique answer to the question Is, a +b+c+d ≥ 50 , the statement is sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step 4, this step is not required
Answer: B
The decimal representation of the number is given above, where z is a positive number and a, b and c are single-digit non-negative integers. Is a = 0?
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find: Is a = 0?
Step 3: Analyze Statement 1 independently
Taking square roots we get
Statement 1 is sufficient to answer the question
Step 4: Analyze Statement 2 independently
Since 4 is positive, squaring both sides twice will not change the sign of inequality:
Statement 2 is sufficient to answer the question
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Steps 3 and 4, this step is not required
Answer: Option D
If |p-5| =3 and |q-3| = 5, which of the following statements must be true?
Given
To Find: The options that must be true(for all values of p and q)
Approach
Working Out
1. As | p – 5| = 3, value of p will be 3 units from away from 5 on the number line. So, following can be values of p:
2. As | q – 3| = 5, value of q will be 5 units away from 3 on the number line. So, following can be values of q:
3. Evaluating Options
Answer: E
How many integrals value of x satisfy the inequality (1-x^{2})(4-x^{2})(9-x^{2}) > 0 ?
Given
To Find: Integral values of x that satisfy the inequality
Approach
As we can see from the wavy line diagram that the inequality is true for the following range:
Thus, there is only 1 integral value of x that satisfy the inequality i.e. 0.
Answer: B
A bag contains red balls that weigh 100 grams each and green balls that weigh 50 grams each. If the number of green balls is 9 more than the number of red balls, how many balls are there in the bag?
(1) If two red balls are added to the bag, the number of red balls will be half the number of green balls
(2) The total weight of the balls in the bag is between 1.05 kilogram and 1.35 kilogram
Steps 1 & 2: Understand Question and Draw Inferences
Let the number of red balls be R and the number of green balls be G.
Note that R and G must be non-negative integers, since these denote the number of balls.
Given: G = R + 9 . . . (I)
And, weight of each Green ball = 50 grams
Weight of each Red ball = 100 grams
Need to find: G + R
Step 3: Analyze Statement 1 independently
Statement 1 says that if two red balls are added to the bag, the number of red balls will be half the number of green balls
R+2=G/2
2R + 4 = G . . . (II)
Equations (I) and (II) form 2 linear equations with 2 unknowns. Sufficient to find R and G.
Step 4: Analyze Statement 2 independently
The total weight of the balls in the bag is between 1.05 kilogram and 1.35 kilogram
1.05 kg = 1050 grams
1.350 kg = 1350 grams
Statement 2 says that
1050< 100R + 50G < 1350
105 <10R+5G < 135
21 < 2R + G < 27
Using (I)
21 < 3R + 9 < 27
7 < R + 3 < 9
4 < R < 6
Sufficient.
Step 5: Analyze Both Statements Together (if needed)
Since we arrive at a unique answer in each of Steps 3 and 4, this step is not required
Answer: D
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