Test: Absolute Values/Modules - GMAT MCQ

A: |x| - 5 < 10
This equation states that the absolute value of x minus 5 is less than 10. However, this does not account for the fact that x is between -5 and 15. For values of x greater than or equal to 5, the equation holds, but for values of x less than 5, it does not. Therefore, option A is not correct.

B: |x| + 5 < 10
Similarly to option A, this equation does not account for the given range of x. For values of x greater than or equal to 0, the equation holds, but for values of x less than 0, it does not. Thus, option B is not correct.

C: |x - 5| < 10
This equation represents the absolute value of x minus 5 being less than 10. It considers the range of x between -5 and 15 since it accounts for both positive and negative values. For any value of x between -5 and 15, the equation holds. Therefore, option C is correct.

D: |x + 5| < 10
This equation represents the absolute value of x plus 5 being less than 10. While it holds for many values of x within the given range, it does not cover the entire range. For values of x less than or equal to -15, the equation does not hold. Thus, option D is not correct.

E: |x + 5| < 0
This equation states that the absolute value of x plus 5 is less than 0. However, the absolute value of any number is always non-negative, so it can never be less than 0. Therefore, option E is not correct.

Based on the analysis, the equation that represents the range of x when x is a number between -5 and 15 is option C: |x - 5| < 10.

Therefore, the correct answer is C.

Case 1: When x is a positive integer
If x is a positive integer, the expression |99 - 7x| can be simplified as 99 - 7x since the absolute value of a positive number is equal to the number itself. In this case, we want to minimize 99 - 7x. The smallest positive integer value for x that still satisfies the given range is x = 14. Plugging this value into the expression, we get |99 - 7(14)| = |99 - 98| = 1.

Case 2: When x is a negative integer
If x is a negative integer, the expression |99 - 7x| can be simplified as -(99 - 7x) since the absolute value of a negative number is equal to its positive counterpart. In this case, we want to minimize -(99 - 7x) or -99 + 7x. The smallest negative integer value for x that still satisfies the given range is x = -14. Plugging this value into the expression, we get |99 - 7(-14)| = |99 + 98| = 197.

Comparing the results from both cases, we see that the smallest possible value of |99 - 7x| is 1. Therefore, the correct answer is B: 1.

To determine the range of solutions to the equation |x² - 1| = 0, we need to solve the equation and find the values of x that satisfy it.

The absolute value of any number is non-negative, and it is equal to zero only when the number itself is zero. So, we have:

x² - 1 = 0

Adding 1 to both sides of the equation:

x² = 1

Taking the square root of both sides (considering both positive and negative roots):

x = ±1

Hence, the solutions to the equation are x = 1 and x = -1.

The range of the solutions is the set of values that satisfy the equation, which is {1, -1}. In other words, the range consists of two distinct values.

Therefore, the correct answer is C: 2, indicating that there are two solutions to the equation.

To find the complete solution set for the inequality |x - 3| > 4, we need to consider two cases: when x - 3 is positive and when x - 3 is negative.

Case 1: (x - 3) > 4
In this case, we have x - 3 > 4. Adding 3 to both sides of the inequality, we get x > 7.

Case 2: -(x - 3) > 4
Here, we have -(x - 3) > 4. Multiplying both sides by -1 and reversing the inequality, we get x - 3 < -4. Adding 3 to both sides, we have x < -1.

Combining both cases, the complete solution set is x < -1 or x > 7. This means that x must be less than -1 or greater than 7.

Therefore, the correct answer is E: x < -1, x > 7.

We can split the equation into two cases, one for the positive value inside the absolute value and one for the negative value inside:

Case 1: 2x - 3 = 7
Solving this equation, we have:
2x - 3 = 7
Adding 3 to both sides:
2x = 10
Dividing both sides by 2:
x = 5

Case 2: -(2x - 3) = 7
Solving this equation, we have:
-(2x - 3) = 7
Multiplying both sides by -1 and distributing the negative sign:
-2x + 3 = 7
Subtracting 3 from both sides:
-2x = 4
Dividing both sides by -2 (which flips the inequality direction):
x = -2

Therefore, the solutions to the equation are x = 5 and x = -2.

The range of solutions refers to the difference between the largest and smallest values. In this case, it is the absolute difference between 5 and -2, which is 7. Thus, the range of solutions to the equation |2x - 3| = 7 is 7.

Therefore, the correct answer is D: 7.

Let's substitute n = 1:

|56 - 5(1)| = |56 - 5| = |51| = 51

Let's substitute n = 2:

|56 - 5(2)| = |56 - 10| = |46| = 46

We can see that as we increase the value of n, the absolute value expression decreases.

Let's substitute n = 9:

|56 - 5(9)| = |56 - 45| = |11| = 11

Let's substitute n = 10:

|56 - 5(10)| = |56 - 50| = |6| = 6

Based on these calculations, we see that the possible values of |56 - 5n| when n is a positive integer are 6, 11, 46, and 51.

Among the given answer choices, the only option that matches one of the possible values is B: 9. Therefore, B is a possible value of |56 - 5n|.

Hence, the correct answer is B.

Case 1: When 100 - 7a is positive
If 100 - 7a is positive, the expression |100 - 7a| can be simplified as 100 - 7a since the absolute value of a positive number is equal to the number itself. In this case, we want to minimize 100 - 7a. To achieve the smallest possible value, we need to maximize the value of 7a. The maximum value for 7a that is less than or equal to 100 is 98 (a = 14). Plugging this value into the expression, we get |100 - 7(14)| = |100 - 98| = 2.

Case 2: When 100 - 7a is negative
If 100 - 7a is negative, the expression |100 - 7a| can be simplified as -(100 - 7a) since the absolute value of a negative number is equal to its positive counterpart. In this case, we want to minimize -(100 - 7a) or -100 + 7a. To achieve the smallest possible value, we need to minimize the value of 7a. The minimum value for 7a that is greater than or equal to 100 is 105 (a = 15). Plugging this value into the expression, we get |100 - 7(15)| = |100 - 105| = 5.

Comparing the results from both cases, we find that the least possible value of |100 - 7a| is 2. Therefore, the correct answer is B: 2.

A: |x - 2| < 4
If we simplify this inequality, we get -4 < x - 2 < 4. Adding 2 to all parts of the inequality, we have -2 < x < 6. This is not equivalent to the given range of -2 < x < 4. So, option A is not correct.

B: |x - 1| < 3
Simplifying this inequality, we get -3 < x - 1 < 3. Adding 1 to all parts of the inequality, we have -2 < x < 4. This is exactly the same as the given range -2 < x < 4. Therefore, option B is correct.

C: |x + 1| < 3
Simplifying this inequality, we get -3 < x + 1 < 3. Subtracting 1 from all parts of the inequality, we have -4 < x < 2. This range is not equivalent to the given range -2 < x < 4. Hence, option C is not correct.

D: |x + 2| < 4
Simplifying this inequality, we get -4 < x + 2 < 4. Subtracting 2 from all parts of the inequality, we have -6 < x < 2. This range is not equivalent to -2 < x < 4. Therefore, option D is not correct.

Based on the analysis, the only option that is equivalent to -2 < x < 4 is option B: |x - 1| < 3.

To determine which statement must be true when y + |y| = 0, let's analyze the given equation.

We know that |y| represents the absolute value of y, which is always non-negative. Therefore, |y| is greater than or equal to 0.

If we have y + |y| = 0, this means that the sum of y and a non-negative number (|y|) is equal to zero. For the sum to be zero, y must be a negative number such that its magnitude (absolute value) is equal to the non-negative value |y|.

From the equation, it is evident that y must be less than or equal to zero (y ≤ 0) to satisfy the condition y + |y| = 0.

Hence, the correct answer is D: y ≤ 0.

Let's analyze each option:

A: n + 1
If we substitute n = -0.5, we get -0.5 + 1 = 0.5.

B: n/2
For n = -0.5, we have -0.5/2 = -0.25.

C: n²
Using n = -0.5, we get (-0.5)² = 0.25.

D: 1/n²
When n = -0.5, we have 1/(-0.5)² = 4.

E: 1/n
Substituting n = -0.5, we get 1/(-0.5) = -2.

Comparing the results, we find that option D: 1/n² yields the greatest absolute value with a result of 4. Therefore, the correct answer is D.