Test: Inequalities - GMAT MCQ

1. 2 - 3x > -10:
Add 3x to both sides: 2 - 3x + 3x > -10 + 3x
Simplify: 2 > -10 + 3x
Add 10 to both sides: 2 + 10 > -10 + 10 + 3x
Simplify: 12 > 3x
Divide both sides by 3 (note that dividing by a negative number reverses the inequality): 12/3 > 3x/3
Simplify: 4 > x

From this inequality, we can conclude that x must be less than 4 in order for 2 - 3x to be greater than -10.

2. 6 - 8x < 14:
Subtract 6 from both sides: 6 - 8x - 6 < 14 - 6
Simplify: -8x < 8
Divide both sides by -8 (remember to reverse the inequality since we're dividing by a negative number): (-8x)/-8 > 8/-8
Simplify: x > -1

From this inequality, we can conclude that x must be greater than -1 for 6 - 8x to be less than 14.

Now let's analyze the given answer choices:

I. -1: This value satisfies both inequalities. From the first inequality, 4 > x, we can see that x can be -1 or any value less than 4. From the second inequality, x > -1, we can see that x can be -1 or any value greater than -1. So -1 can be a valid value for x.

II. 0: This value satisfies both inequalities. From the first inequality, 4 > x, we can see that x can be 0 or any value less than 4. From the second inequality, x > -1, we can see that x can be 0 or any value greater than -1. So 0 can be a valid value for x.

III. 4: This value does not satisfy the first inequality, 4 > x. If x = 4, the inequality becomes 4 > 4, which is not true. Therefore, 4 cannot be a valid value for x.

Based on our analysis, only III (4) cannot be the value of x. So the correct answer is C.

To determine which of the given statements must be true when x < 0, let's examine each statement individually:

I. x² > 0:
For any real number x, squaring it will always result in a non-negative value (greater than or equal to zero). However, since x < 0 in this case, x^2 will be positive. Therefore, statement I is true.

II. x − 2x > 0:
Simplifying the expression, we get -x > 0. Since x < 0, multiplying both sides of the inequality by -1 flips the sign, giving us x < 0, which is true. Therefore, statement II is true.

III. x³ + x² < 0:
Substituting x = -1 into the expression, we get (-1)³ + (-1)² = -1 + 1 = 0. Since the expression evaluates to zero, it is not less than zero. Therefore, statement III is false.

Based on our analysis, statements I and II are true, while statement III is false. Thus, the correct answer is (B) I & II.

To find the area of the region that satisfies the given inequalities, we first need to graph the inequalities on a coordinate plane.

The first inequality, x + y - 3 ≤ 0, can be rearranged to y ≤ -x + 3. This represents a line with a y-intercept of 3 and a slope of -1.

The second inequality, x ≥ 0, represents the positive x-axis.

The third inequality, y ≥ 0, represents the positive y-axis.

To determine the area of the region that satisfies all the inequalities, we need to identify the enclosed region on the graph.

Since x ≥ 0 and y ≥ 0, we know that the region is bounded by the positive x-axis and the positive y-axis.

The line y ≤ -x + 3 intersects the x-axis at (3, 0) and the y-axis at (0, 3).

To find the area of the region, we can calculate the area of the triangle formed by the points (0, 0), (3, 0), and (0, 3).

The base of the triangle is the line segment from (0, 0) to (3, 0), which has a length of 3.

The height of the triangle is the line segment from (0, 0) to (0, 3), which has a length of 3.

Therefore, the area of the triangle is (base * height) / 2 = (3 * 3) / 2 = 9 / 2 = 4.5.

Therefore, the correct answer is D. 4.5.

Given:
1. 5 ≥ a > 1
2. b ≥ –2
To find the maximum value of a – b, we should choose the largest possible value for a and the smallest possible value for b.

Max value of a: Since 5 ≥ a > 1, the largest possible value for a is 5.

Min value of b: Since b ≥ –2, the smallest possible value for b is –2.

Therefore, the maximum value of a – b is 5 - (-2) = 7.

To find the minimum value of a – b, we should choose the smallest possible value for a and the largest possible value for b.

Min value of a: Since 5 ≥ a > 1, the smallest possible value for a is 2.

Max value of b: Since b ≥ –2, there is no restriction on the maximum value of b.

Therefore, the minimum value of a – b is 2 - (-2) = 4.

So, the range of possible values for a – b is 4 ≤ a – b ≤ 7.

Now let's analyze the given answer choices:

A. –5: This value is within the range of possible values (4 ≤ a – b ≤ 7).
B. –3: This value is within the range of possible values (4 ≤ a – b ≤ 7).
C. 2: This value is within the range of possible values (4 ≤ a – b ≤ 7).
D. 7: This value is within the range of possible values (4 ≤ a – b ≤ 7).
E. 8: This value is NOT within the range of possible values (4 ≤ a – b ≤ 7).

Therefore, the answer is E. The value 8 cannot be the result of a – b based on the given conditions.

To find the average value of a, b, c, and 29, we can start by calculating the sum of these numbers and then divide it by the total count of numbers.

We are given that the average of a, b, c, 14, and 15 is 12. This can be expressed as:

(a + b + c + 14 + 15) / 5 = 12

Now, we can simplify this equation:

(a + b + c + 29) / 4 = ?

To find the average value of a, b, c, and 29, we need to find the sum of these numbers. Rearranging the equation above, we have:

(a + b + c + 29) = 4 * ?

Since we don't know the value of ?, let's solve for it. We can substitute the given average of 12 into the first equation:

(a + b + c + 14 + 15) / 5 = 12

(a + b + c + 29) = 12 * 5

(a + b + c + 29) = 60

Now we know that the sum of a, b, c, and 29 is 60. Substituting this value into the second equation:

(a + b + c + 29) = 4 * ?

60 = 4 * ?

To solve for ?, we divide both sides by 4:

? = 60 / 4

? = 15

Therefore, the value of ? is 15. This means the average value of a, b, c, and 29 is 15. Therefore, the correct answer is option D.

To find the least integer n such that 1/2ⁿ is less than 0.001, we need to solve the inequality:

1/2^n < 0.001

To simplify the inequality, we can multiply both sides by 2^n to get:

1 < 0.001 * 2ⁿ

Now, we can simplify the right side:

1 < 0.001 * 2ⁿ 1 < 0.001 * (2ⁿ) 1 < 0.001 * 2ⁿ 1 < (2/1000) * 2ⁿ 1 < (2ⁿ)/1000

Now, we can rearrange the inequality to isolate 2ⁿ:

1000 < 2ⁿ

To determine the least integer n that satisfies this inequality, we can take the logarithm base 2 of both sides:

log2(1000) < log2(2ⁿ)

Simplifying further:

log2(1000) < n

Using a calculator, we can find that log2(1000) ≈ 9.966. Therefore, we have:

9.966 < n

Since n must be an integer, the smallest integer greater than 9.966 is 10. Hence, the least integer n that satisfies the inequality 1/2ⁿ < 0.001 is 10.

Therefore, the correct answer is A) 10.

To find the possible integer values for x in the given inequality, let's break it down into two cases based on the absolute value expression:

Case 1: (4x - 3) < 6 Solving this inequality, we get: 4x - 3 < 6 Adding 3 to both sides: 4x < 9 Dividing both sides by 4 (since the coefficient of x is positive): x < 9/4

Since we are looking for integer values of x, the possible values for x in this case are 1, 2.

Case 2: -(4x - 3) < 6 Simplifying the inequality by distributing the negative sign: -4x + 3 < 6 Subtracting 3 from both sides: -4x < 3 Dividing both sides by -4 (since the coefficient of x is negative, we need to reverse the inequality sign): x > 3/(-4) x > -3/4

Again, considering integer values of x, the possible values in this case are 0, -1, -2.

Combining the results from both cases, we have the following possible integer values for x: -2, -1, 0, 1, 2.

Therefore, there are a total of 5 possible integer values for x, and the correct answer is E. Five.

To solve this inequality, we can break it down into two cases:

Case 1: x is positive or zero (x ≥ 0) If x is greater than or equal to 0, the inequality simplifies to: x - 8 > x + 4

By subtracting x from both sides, we get: -8 > 4

This inequality is false, so there are no solutions for x in this case.

Case 2: x is negative (x < 0) If x is less than 0, the inequality simplifies to: -(x - 8) > x + 4

Expanding the absolute values, we have: -1(x - 8) > x + 4

Simplifying further:

• x + 8 > x + 4

By subtracting x from both sides, we get: 8 > 2x + 4

Subtracting 4 from both sides: 4 > 2x

Dividing by 2: 2 > x

So, in this case, x must be less than 2.

Combining the results from both cases, we find that x must be less than 2. However, we also have the constraint that |x| < 20, which means x must be within the range -20 < x < 20.

Combining these conditions, the allowable range for x is -20 < x < 2, which corresponds to option (B).

The given inequality is:
x³ < x²

To understand this inequality, let's consider the following cases:

Case 1: x > 0
If x is a positive number, raising it to any positive power will yield a positive result. Therefore, x³ and x² will both be positive. However, since x³ is less than x² in this case, we can conclude that x³ - x² < 0.

Case 2: x = 0
If x is equal to zero, both x³ and x² will be zero. In this case, x³ - x² = 0 - 0 = 0.

Case 3: x < 0
If x is a negative number, raising it to an odd power will result in a negative number, while raising it to an even power will yield a positive number. Therefore, x³ will be negative, and x² will be positive. Since x³ is less than x² in this case, we can conclude that x³ - x² < 0.

Based on the analysis of the inequality, we can see that x³ - x² < 0 for all values of x except when x equals zero. Now, let's examine the options:

A. x: This option does not necessarily have to be negative since x can be positive.
B. -x: This option does not necessarily have to be negative either since x can be positive.
C. x⁵: This option does not have to be negative either. It can be positive or negative depending on the value of x.
D. x - 1: This option must be negative. Since we know that x³ - x² < 0, we can rewrite the inequality as x³ < x².
By subtracting x² from both sides, we get x³ - x² < 0. Factoring out an x, we have x(x² - 1) < 0. Solving the equation x² - 1 = 0, we find that x = -1 or x = 1. Therefore, for x < -1 or -1 < x < 1, x(x² - 1) < 0. This means that x - 1 is negative for x < -1 or -1 < x < 1.

E. x^(-1): This option does not necessarily have to be negative either. It can be positive or negative depending on the value of x.

In conclusion, the option that must be negative is D. x - 1.

I. x is negative: If x were positive, then |x/y| would be equal to x/y, and since x/y < |x/y|, this would contradict the given condition. Therefore, x must be negative. Hence, statement I is true.

II. y is positive: If y were negative, then |x/y| would be equal to -x/y, and since x/y < |x/y|, this would also contradict the given condition. Therefore, y must be positive. Hence, statement II is true.

III. x/y is negative: If x/y were positive, then |x/y| would be equal to x/y, and since x/y < |x/y|, this would once again contradict the given condition. Therefore, x/y must be negative. Hence, statement III is true.

Based on the analysis above, all three statements are true. Therefore, the answer is option (E) "I, II, and III."