Test: Inequalities - GMAT MCQ

To solve the inequality |x - 5| ≤ 2.5, we can consider two cases:

Case 1: x - 5 ≥ 0
In this case, the absolute value |x - 5| is equal to x - 5.
Substituting this into the inequality, we have x - 5 ≤ 2.5.
By adding 5 to both sides of the inequality, we get x ≤ 7.5.

Case 2: x - 5 < 0
In this case, the absolute value |x - 5| is equal to -(x - 5) = -x + 5.
Substituting this into the inequality, we have -x + 5 ≤ 2.5.
By subtracting 5 from both sides of the inequality and multiplying by -1, we get x ≥ 2.5.

Combining the results from both cases, we have 2.5 ≤ x ≤ 7.5.

To find the number of integer values of x that satisfy the inequality, we can count the integers within this range.

The integers that satisfy the inequality are 3, 4, 5, 6, and 7. (Note that 2.5 and 7.5 are not included since the inequality is not inclusive.)

Therefore, there are 5 integer values of x that satisfy the inequality.

Thus, the correct answer is C: 5.

Let's analyze each statement individually:

I. x³ < x²:
Since x is negative and |x| is decreasing as x approaches 0, x³ will be smaller than x². Therefore, statement I is true.

II. x⁵ < 1 - x:
Since x is negative, x^5 will be negative. On the right side, 1 - x is positive because x is negative. Therefore, statement II is true.

III. x⁴ < x²:
Since x is negative and |x| is decreasing as x approaches 0, x⁴ will be smaller than x². Therefore, statement III is true.

Based on our analysis, all three statements (I, II, and III) are true. Therefore, the correct answer is:

E: I, II, and III

Given the conditions m < 15 and m > -4, we can determine the range of possible values for m.

Since m is less than 15, we can conclude that m is greater than -15 because -15 is smaller than any value less than 15.

Therefore, the correct statement is:

A: m > -15

This means that m must be greater than -15 based on the given conditions. Therefore, option A is the correct answer.

(x - 1)(x + 3) > 0:
To solve this inequality, we can use the concept of sign charts or test intervals. We look for the intervals where the expression (x - 1)(x + 3) is greater than 0, meaning the expression is positive.
(x - 1)(x + 3) > 0 is positive when both factors have the same sign: either both positive or both negative.

(x - 1) > 0 and (x + 3) > 0:
x > 1 and x > -3
x > 1 (as x > -3 is already satisfied)

So, the first inequality is satisfied when x > 1.

(x + 5)(x - 4) < 0:
To solve this inequality, we can again use the concept of sign charts or test intervals. We look for the intervals where the expression (x + 5)(x - 4) is less than 0, meaning the expression is negative.
(x + 5)(x - 4) < 0 is negative when the factors have different signs: one positive and one negative.

(x + 5) < 0 and (x - 4) > 0:
x < -5 and x > 4
-5 < x < 4

So, the second inequality is satisfied when -5 < x < 4.

Now, let's check which values of x satisfy both inequalities:

-5 < x < 4 and x > 1

The only values that satisfy both inequalities are:

II. -4 (satisfies both inequalities)
III. 2 (satisfies both inequalities)

Therefore, the correct answer is C: II and III only.

To solve the absolute value inequality |4n - 3| ≤ 2, we can consider two cases:

Case 1: 4n - 3 ≥ 0
If 4n - 3 ≥ 0, then |4n - 3| = 4n - 3. In this case, the inequality becomes 4n - 3 ≤ 2, which we can solve as follows:

4n - 3 ≤ 2
4n ≤ 5
n ≤ 5/4

Case 2: 4n - 3 < 0
If 4n - 3 < 0, then |4n - 3| = -(4n - 3) = -4n + 3. In this case, the inequality becomes -4n + 3 ≤ 2, which we can solve as follows:

-4n + 3 ≤ 2
-4n ≤ -1
n ≥ 1/4

Now, we can combine the results from both cases. Since we want to find the integers that satisfy the inequality, we need to find the intersection of the solutions from both cases.

The solutions from Case 1 are n ≤ 5/4, and the solutions from Case 2 are n ≥ 1/4.

The only integer that satisfies both conditions is n = 1.

Therefore, the correct answer is that only one integer, n = 1, satisfies the absolute value inequality.

Hence, the answer is A.

To solve this problem, we need to find the circumference of the circular jogging track, which is also the distance Johanna walked. The circumference of a circle is given by the formula C = πd, where C is the circumference and d is the diameter.

Given that the diameter of the lake is 2 miles, the radius is half of the diameter, which is 1 mile. Therefore, the circumference of the track is C = π(1) = π miles.

Johanna walked once around the track at an average speed of 3 miles per hour. We can use the formula speed = distance/time to find the time it took her to walk completely around the lake.

3 miles/hour = π miles / t hours

To solve for t, we divide both sides of the equation by π:

3 / π = t

Now, we need to determine the value of t. We can approximate π as 3.14.

3 / 3.14 ≈ 0.955

Therefore, the correct answer is that 2.0 < t < 2.5, which corresponds to option C.

Given that x + y = 5, we can analyze the options:

A: x and y are consecutive integers.
The statement does not provide any information about the consecutive nature of x and y. For example, x could be 2 and y could be 3, which are consecutive, or x could be 1 and y could be 4, which are not consecutive. Therefore, option A is not necessarily true.

B: If x < 0, then y > 0.
Since x + y = 5, if x is negative, then y must be positive in order for their sum to be positive. Therefore, if x < 0, then y > 0 is true.

C: If x > 0, then y < 0.
There is no information provided in the statement that establishes a relationship between the signs of x and y. Therefore, we cannot determine if if x > 0, then y < 0 is true or not.

D: Both x and y are even.
There is no information provided in the statement that establishes a relationship between the parity of x and y. Therefore, we cannot determine if both x and y are even or not.

E: Both x and y are less than 5.
Since x + y = 5, it is not necessary for both x and y to be less than 5. For example, x could be 6 and y could be -1, which sums to 5 but includes a number greater than 5. Therefore, both x and y are less than 5 is not necessarily true.

Therefore, the correct answer is B: If x < 0, then y > 0.

Given that a > b and c > d, we can analyze the options:

A: a - b > c + d
We cannot determine the relationship between the differences a - b and c + d based solely on the given information. Therefore, option A cannot be determined.

B: a - c > b - d
We cannot determine the relationship between a - c and b - d based solely on the given information. Therefore, option B cannot be determined.

C: c + d < a - b
Since a > b, subtracting b from a will always result in a positive value. Similarly, since c > d, adding d to c will also result in a positive value. Therefore, c + d < a - b is not true.

D: b + d < a + c
Since a > b and c > d, the sum of a and c will always be greater than the sum of b and d. Therefore, b + d < a + c is true.

E: a - c < b + d
We cannot determine the relationship between a - c and b + d based solely on the given information. Therefore, option E cannot be determined.

Therefore, the correct answer is D: b + d < a + c.

Given that a < 0 and b < c, we can analyze the options:

A: ab < c
Since a < 0, multiplying a negative number by b results in a positive product. However, the statement does not specify whether c is positive or negative. Therefore, we cannot determine if ab < c is true or not.

B: ac > b
This statement involves the product of a and c, but we do not have any information about the relationship between a and c. Therefore, we cannot determine if ac > b is true or not.

C: ab > 0
Since a < 0, multiplying a negative number by any value of b will always result in a negative product. Therefore, ab > 0 is not true.

D: ac < 0
Since a < 0, and c could be positive or negative, multiplying a negative number (a) by any value of c can result in a positive or negative product. Therefore, ac < 0 is not necessarily true.

E: ab > ac
Since a < 0, multiplying a negative number by b will always result in a negative product. Similarly, multiplying a negative number by c will also result in a negative product. Since negative numbers multiplied together yield a positive product, ab > ac must be true.

Therefore, the correct answer is E: ab > ac.