Test: Absolute Values/Modules - GMAT MCQ

To determine if z > 2, let's evaluate each statement separately:

Statement (1): |(z-2)²| > 4
Squaring both sides of the equation, we get: (z-2)² > 4
Taking the square root of both sides, we have: |z-2| > 2

This statement tells us that the absolute value of z minus 2 is greater than 2. In other words, z can be either less than 0 or greater than 4. It does not provide a definite answer to whether z > 2. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): |8x - 4k| = z
This statement provides an equation relating z to the variables x and k. Without any additional information about x and k, we cannot determine the value of z or whether it is greater than 2. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together:
From statement (1), we know that |z-2| > 2, which implies z can be less than 0 or greater than 4.
From statement (2), we have the equation |8x - 4k| = z. Without specific values for x and k, we cannot determine the value of z.

Combining both statements, we still cannot determine a definite answer to whether z > 2. Therefore, both statements together are not sufficient to answer the question.

Hence, the correct answer is (C) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement 1: 4 < x < 6

=> x can take any value between 4 and 6 and we cannot obtain a specific value using Statement 1 alone.

Statement 2: |x| = 4x − 15

=> x = 4x -15 OR x = -(4x-15) = -4x +15

=> x = 5 => x=3

At x = 3 we have |x| = 4x - 15

= 4*3 - 15

=> |x| = -3

At x=5, we have |x| = 4*5 - 15

=> +5

A modulus value cannot yield negative result and hence x=3 cannot be considered a value for x.
Thus x = 5
Hence OPTION(B)

We can solve the question using two approaches -
Approach 1: Using the concept of absolute value
x² > y²
Taking square root on both sides
|x| > |y|
Inference: Is the distance of x from 0 greater than the distance of y from 0

Statement 1
|x| > y
This statement tells us that the distance of x from 0 is greater than the value of y. However, we do not know anything about the distance of y from 0.
Case 1: ---------- 0 ----- y ---------- x -----
Is the distance of x from 0 greater than the distance of y from 0 - Yes !
Case 2: ---------- y --------------------- 0 -- x -----
Is the distance of x from 0 greater than the distance of y from 0 - No !
The statement is not sufficient. Hence we can eliminate A and D.
Statement 2
x>|y|
The value of x is greater than the distance of y from 0. As modulus (the distance) is always non negative, we can conclude that x will be positive. There can be two cases
Case 1: ---------- y ------- 0 -------------------------- x -----------
Case 2: ---------- 0 ------- y -------------------------- x -----------
In both cases, Is the distance of x from 0 greater than the distance of y from 0 - Yes !
Sufficient

Approach 2: Picking Numbers

Statement 1
|x| > y
Case 1: x = 10 ; y = 2
Is x² > y² -- Yes !
Case 2: x = 10 ; y = -12
Is x² > y² -- No!
The statement is not sufficient. Hence we can eliminate A and D.
Statement 2
|y| is non negative, hence x is positive.
Case 1: x = 10 ; y = 2
Is x² > y² -- Yes !
Case 2: x = 10 ; y = -9
Is x² > y² -- Yes !

To analyze the given question, let's consider each statement separately:

Statement (1): a/b < 0

This statement tells us that the ratio of a to b is negative. This means that either a and b have opposite signs (one is positive, and the other is negative), or one of them is zero. In either case, we can conclude that |a - b| < |a| + |b|.

If a and b have opposite signs, then the absolute value of their difference, |a - b|, will always be greater than the sum of their absolute values, |a| + |b|. For example, if a = 5 and b = -3, we have |a - b| = |5 - (-3)| = |8| = 8, and |a| + |b| = |5| + |-3| = 8. Since 8 is not less than 8, the inequality holds.

If either a or b is zero, then the left side of the inequality, |a - b|, will be equal to the absolute value of the non-zero number, and the right side, |a| + |b|, will be equal to the absolute value of the non-zero number plus zero. Since the absolute value of a non-zero number is always greater than or equal to zero, the inequality |a - b| < |a| + |b| also holds in this case.

Statement (2): a² * b < 0

This statement tells us that the product of a² and b is negative. From this statement alone, we cannot determine the signs or values of a and b individually, so we cannot make a conclusive decision about the inequality |a - b| < |a| + |b|. Therefore, statement (2) alone is not sufficient to answer the question.

Combining the two statements:

When we consider both statements together, we can determine that a/b < 0 and a^2b < 0 simultaneously. This implies that both a/b and a^2b have opposite signs. Since a²*b < 0, we know that a and b have opposite signs. This information allows us to conclude that |a - b| < |a| + |b|.

Therefore, the correct answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement 1

x^a ∗ x^b = 1
x^{(a + b)} = 1
Case 1: x = -1
a + b must be even
Case 2: x = 1
a + b can be even or a + b can be odd
Case 3: x = 0
a + b can be even, except 0 or a + b can be odd
As we have different answers corresponding to different value of x, the statement is not sufficient. Eliminate A and D

Statement 2
a ≠ −1
The statement is not sufficient. Eliminate B.
Combined
The statements combined is still not sufficient as in Case 2 and Case 3 a + b can be even or a + b can be odd.

Statement (1): |x + 1| = 2|x - 1|
This statement gives us an equation relating the absolute values of x + 1 and x - 1. To solve this equation, we can consider two cases:

Case 1: x + 1 ≥ 0 and x - 1 ≥ 0
In this case, both expressions inside the absolute values are positive, so we can remove the absolute value signs:
x + 1 = 2(x - 1)
x + 1 = 2x - 2
x = 3

Case 2: x + 1 < 0 and x - 1 < 0
In this case, both expressions inside the absolute values are negative, so we need to change the signs:
-(x + 1) = 2-(x - 1)
-x - 1 = -2x + 2
x = -3

From both cases, we have two possible values for x: x = 3 and x = -3.

Statement (2): |x - 3| ≠ 0
This statement tells us that the absolute value of x - 3 is not equal to zero. This means that x - 3 cannot be zero, which implies x ≠ 3.

Combining both statements, we have x = 3 and x ≠ 3. Therefore, the only value that satisfies both conditions is x = -3. Since -3 is less than 1, we can conclude that |x| < 1.

Hence, both statements together are sufficient to answer the question, but neither statement alone is sufficient. Thus, the correct answer is (C).

Let us start be examining the conditions necessary for |a|b > 0. Since |a| cannot be negative, both |a| and b must be positive. However, since |a| is positive whether a is negative or positive, the only condition for a is that it must be non-zero.
Hence, the question can be restated in terms of the necessary conditions for it to be answered "yes":
“Do both of the following conditions exist: a is non-zero AND b is positive?”
(1) INSUFFICIENT: In order for a = 0, |a^b| would have to equal 0 since 0 raised to any power is always 0. Therefore (1) implies that a is non-zero. However, given that a is non-zero, b can be anything for |ab| > 0 so we cannot determine the sign of b.

(2) INSUFFICIENT: If a = 0, |a| = 0, and |a|^b = 0 for any b. Hence, a must be non-zero and the first condition (a is not equal to 0) of the restated question is met. We now need to test whether the second condition is met. (Note: If a had been zero, we would have been able to conclude right away that (2) is sufficient because we would answer "no" to the question is |a|b > 0?) Given that a is non-zero, |a| must be positive integer. At first glance, it seems that b must be positive because a positive integer raised to a negative integer is typically fractional (e.g., a -2 = 1/a². Hence, it appears that b cannot be negative. However, there is a special case where this is not so:

If |a| = 1, then b could be anything (positive, negative, or zero) since  |1|^b is always equal to 1, which is a non-zero integer . In addition, there is also the possibility that b = 0. If |b| = 0, then |a|⁰ is always 1, which is a non-zero integer.

Hence, based on (2) alone, we cannot determine whether b is positive and we cannot answer the question.

An alternative way to analyze this (or to confirm the above) is to create a chart using simple numbers as follows:

We can quickly confirm that (2) alone does not provide enough information to answer the question.

(1) AND (2) INSUFFICIENT: The analysis for (2) shows that (2) is insufficient because, while we can conclude that a is non-zero, we cannot determine whether b is positive. (1) also implies that a is non-zero, but does not provide any information about b other than that it could be anything. Consequently, (1) does not add any information to (2) regarding b to help answer the question and (1) and (2) together are still insufficient. (Note: As a quick check, the above chart can also be used to analyze (1) and (2) together since all of the values in column 1 are also consistent with (1)).

The correct answer is E.

If a = |b - 6| + |b + 2|, what is the value of a ?
critical points are 6 and -2. so we have 3 regions for b : < -2 ; -2 to 6 ; > 6
(1) a is an integer greater than 7
lets plugin 3 different values one from each region
b = -3; a = |-3 - 6| + |-3 + 2| = 10
b = 0; a = |0 - 6| + |0 + 2| = 8
b = 8; a = |8 - 6| + |8 + 2| = 12
Since we get different values of 'a' in each region statement 1 is Not sufficient

(2) -2 < b < 6
b = -1; a = |-1 - 6| + |-1 + 2| = 8
b = 0; a = |0 - 6| + |0 + 2| = 8
b = 4; a = |4 - 6| + |4 + 2| = 8
'a' is same for the given range of b, hence, statement 2 is sufficient

To find the value of a - b, we need to determine the values of both a and b individually. Let's analyze the given information and the statements one by one:

Statement (1): a³ * b⁷ > 0

From this statement, we can deduce that both a³ and b⁷ have the same sign. Since the product of two positive numbers is positive and the product of two negative numbers is also positive, we can conclude that a³ and b⁷ are both positive or both negative. This information alone does not provide us with any specific values for a or b. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): a + b > 0

This statement tells us that the sum of a and b is greater than zero. It does not give us any information about their individual values. For example, a could be a positive number and b a negative number, or vice versa, as long as their sum is positive. Thus, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we can combine the information:

From statement (1), we know that a³ and b⁷ have the same sign.

From statement (2), we know that a + b > 0, which implies that a and b have the same sign (either both positive or both negative).

Combining these two pieces of information, we can conclude that a, b, a³, and b⁷ all have the same sign.

Now, let's consider the equation a⁶ = b³ = |x|/x.

Since |x|/x is either 1 or -1 (depending on the sign of x), we can conclude that a⁶ = b³ is either 1 or -1.

If a⁶ = b³ = 1, it implies that both a and b must be 1. Therefore, a - b = 1 - 1 = 0.

If a⁶ = b³ = -1, it implies that both a and b must be -1. Therefore, a - b = -1 - (-1) = 0.

In both cases, we find that a - b = 0.

Therefore, when we consider both statements together, we can determine that the value of a - b is always 0. Hence, the correct solution is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

To determine whether xy > y², let's evaluate each statement separately:

Statement (1): yx < 1
This statement provides information about the relative magnitudes of y and x but does not directly compare xy and y². Without further information, we cannot determine the relationship between xy and y². Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): |x| > y
This statement gives us a comparison between the absolute value of x and y. However, it doesn't provide any specific information about the relationship between xy and y². Like statement (1), statement (2) alone is also not sufficient to answer the question.

Considering both statements together:
When we consider both statements together, we still cannot determine the relationship between xy and y². Even though we have additional information about the magnitudes of x and y, we don't have enough information to establish a definitive relationship between xy and y². Therefore, both statements together are not sufficient to answer the question.

Since neither statement alone nor both statements together are sufficient, the correct answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.