Test: Absolute Values/Modules - GMAT MCQ

We can eliminate options C and D right away: |3−x| = |x−3|, so these two options will have the same maximum value and since we cannot have two correct answers in PS questions then none of them is correct.
Evaluate each option by plugging min and max possible values of x:
A. 3x – 1 → max for x = -2 → |3*(-2)-1| = 7.

B. x² + 1 → max for x = -2 or x=2 → |2² + 1| = 5.

x² – x → max for x = -2 → |(-2)² - (-2)| = 6.

A = - |B|

means A = B or A = -B

A - B = -10 --------(I)

Replace A with B,

B - B = -10

0 = -10 (Not true)

Replace A with -B,

-B -B = -10

B = 5

Hence (A)

||x−3| − 2| = 1| 

⇒ |x−3|−2 = 1 or |x−3|−2 = −1 
⇒ |x−3| = 3 or |x−3| = 1 
⇒ x − 3 = 3 or x−3 =−3 or x−3 = 1 or x−3 = −1 
Therefore x = 6 or 0 or 4 or 2. Hence answer is 4 .

• We have to plug different values of k and get 17k as close to 129 as possible
• If k = 7, then 17k = 17 × 7 = 119 and |129 − 17k| = |129−119| = |10| = 10
• But if k = 8, then 17k = 17 × 8 = 136  and |129 − 17k| = |129−136| = |−7| = 7

To determine the number of possible values of m that satisfy the inequality |m + 1| - |m - 3| > 4, we can simplify the expression by considering different cases.

Case 1: m < -1
In this case, both m + 1 and m - 3 are negative. Thus, the inequality becomes:
-(m + 1) - (-(m - 3)) > 4
m - 1 + m - 3 > 4
-4 > 4
This inequality is not satisfied, so there are no solutions in this case.
Case 2: -1 ≤ m < 3
In this case, m + 1 is non-negative, and m - 3 is negative. The inequality becomes:
(m + 1) - (-(m - 3)) > 4
m + 1 + m - 3 > 4
2m - 2 > 4
2m > 6
m > 3
Since m must be an integer in this case, there are no values of m that satisfy this inequality.

Case 3: m ≥ 3
In this case, both m + 1 and m - 3 are non-negative. The inequality becomes:
(m + 1) - (m - 3) > 4
m + 1 - m + 3 > 4
4 > 4
This inequality is not satisfied, so there are no solutions in this case.

Based on the analysis of all cases, there are no possible values of m that satisfy the inequality.

Therefore, the correct answer is A: 0.

Given that x < y < 0, where x and y are negative:

I. The inequality |x| > |y| will always be true. This can be verified by plugging in values such as -2 < -1 < 0 or -0.50 < -0.25 < 0, where the absolute value of x is greater than the absolute value of y.

II. The inequality x/y > 1 holds true. In this case, y will always be less than x, for example, -4/-2 > 1, -0.1/-0.05 > 1, -6/-5 > 1. Thus, the ratio x/y is always greater than 1.

At this point, we can confidently mark option D as the correct answer.

III. The inequality x^y < 0 is not necessarily true. If y is an even power, the relationship becomes positive, which is not less than 0. If y is an odd power, the relationship becomes negative, satisfying the condition of being less than 0.

Given:
|x|/|3| > 1

We can simplify this inequality by multiplying both sides by |3|, which is a positive value:

|x| > |3|

Option a) x > 3:
This is not true in all cases because x could also be less than -3. For example, x = -4 satisfies |x| > 3 (since |-4| = 4), but it is less than 3, so this is not the only possible solution.

Option b) x < 3:
This is incorrect because x can also be greater than 3. For example, x = 4 also satisfies |x| > 3, but x is greater than 3. So, x is not limited to being less than 3.

Option c) x = 3:
This is incorrect because the inequality says |x| > 3, meaning x cannot be equal to 3. So, x = 3 does not satisfy the inequality.

Option d) x ≠ 3:
This is correct because, based on |x| > 3, x must be either greater than 3 or less than -3, which means x cannot equal 3.

Option e) x < -3:
This is partly true, but not the full solution. x could be less than -3, but it could also be greater than 3. For example, x = 4 also satisfies the inequality. So, this option doesn't cover all possibilities.

Conclusion:

The correct answer is d) x ≠ 3, because the inequality requires |x| > 3, which means x cannot be exactly 3.

To find the condition that must be true for a negative value of x, given that |1/x| > 50, we can analyze the options.

A: x < -50
This option does not necessarily hold true since we don't have specific information about the magnitude of x.

B: x < -10
Similar to option A, we don't have enough information to determine if this option is true.

C: -1 < x < -1/2
This option does not align with the given condition because x is stated to be negative, and this option includes values greater than -1.

D: -1/50 < x < 0
This option satisfies the given condition since x is negative (less than 0), and the inequality range includes values between -1/50 and 0.

E: x > 1/50
This option contradicts the given condition since x is stated to be negative, but this option specifies positive values.

Therefore, the correct answer is D: -1/50 < x < 0.

Given: |4x - 3| < 6

-6+3 < 4x < 6+3

-3/4 < x < 9/4

-0.75 < x < 2.25

values are 0, 1, 2 (Three integer values)

To find the value of a + |a - 6| when a < 6, we need to consider two cases:

When a < 6, the expression |a - 6| evaluates to 6 - a since a - 6 will be negative.
Adding a to |a - 6| gives a + (6 - a), which simplifies to 6.
Therefore, the value of a + |a - 6| when a < 6 is always 6.

The correct answer is C: 6.