Coded Inequalities -1

Question 1: How many positive integer values can x take that satisfy the inequality (x - 8) (x - 10) (x - 12).......(x - 100) < 0?
A. 25
B. 30
C. 35
D. 40

Answer. 30
Explanation.
There are factors (x - 8), (x - 10), (x - 12), ..., (x - 100). Count the number of factors.
Number of terms = (100 - 8)/2 + 1 = 47.
If x equals any root 8, 10, ..., 100, the product is zero and does not satisfy < 0.
If x > 100, then every factor is positive and the product is positive, so it is not accepted.
If x < 8 then every factor is negative and there are 47 negative factors, so the product is negative; thus positive integers 1 through 7 all work.
Sign of the product changes on crossing each simple root. Starting from (-∞,8), the product is negative, and the intervals between consecutive even roots alternate sign. Negative intervals are (-∞,8), (10,12), (14,16), ..., (98,100).
Integers contained in these later negative intervals are the odd integers 11, 15, 19, ..., 99.
Count of these odd integers from 11 to 99 with common difference 4 is (99 - 11)/4 + 1 = 23.
Total positive integer solutions = 7 (from 1 to 7) + 23 = 30.
Hence the answer is "30"
Choice B is the correct answer.

Question 2: Solve the inequality x³ - 5x² + 8x - 4 > 0.

A. (2, ∞)
B. (1, 2) ∪ (2, ∞)
C. (-∞, 1) ∪ (2, ∞)
D. (-∞, 1)

Answer. (1, 2) ∪ (2, ∞)

Explanation.
The cubic polynomial P(x) = x³ - 5x² + 8x - 4 has coefficients summing to zero, so P(1) = 0 and (x - 1) is a factor.
Divide P(x) by (x - 1) to obtain a quadratic factor.
(x - 1) divides P(x) and the quotient is x² - 4x + 4.
Thus P(x) = (x - 1)(x - 2)(x - 2) = (x - 1)(x - 2)².
Analyze the sign of the product (x - 1)(x - 2)².
The factor (x - 2)² is always ≥ 0 and is zero exactly at x = 2; it does not change sign across x = 2.
The sign of P(x) is determined by (x - 1) except that zeros at x = 2 must be excluded because the inequality is strict (> 0).
For x < 1 the product is negative; for 1 < x < 2 the product is positive; for x > 2 the product is positive.
Therefore the solution set is (1, 2) ∪ (2, ∞).
Hence the answer is "(1, 2) ∪ (2, ∞)"
Choice B is the correct answer.

Question 3: Find the range of x for which (x + 2) (x + 5) > 40?

Answer. x < -10 or x > 3

Explanation.
Expand and rearrange.
(x + 2)(x + 5) > 40 ⇒ x² + 7x + 10 > 40.
Subtract 40 from both sides.
x² + 7x - 30 > 0.
Factorize the quadratic.
(x + 10)(x - 3) > 0.
Roots are x = -10 and x = 3. For a upward-opening quadratic the expression > 0 outside the roots.
Hence x < -10 or x > 3.
Alternate quick check: For the product of two numbers to exceed 40, either both are greater than the positive factors 5 and 8 shifted appropriately or both less than corresponding negatives. The algebraic method above is robust.
Hence the answer is "x < -10 or x > 3"

Question 4: How many integer values of x satisfy the inequality x( x + 2)(x + 4)(x + 6) < 200?

Explanation.
Check zeros and sign behaviour.
Zeros occur at x = 0, -2, -4, -6; at these the product is 0 and satisfies < 200.
There are four factors. If all four factors are positive or all four are negative the product is positive. If exactly one or exactly three factors are negative the product is negative and therefore < 200.
Check integers near zero.
x = -1 gives product -1·1·3·5 = -15 < 200.
x = 1 gives 1·3·5·7 = 105 < 200.
x = -7 gives -7·-5·-3·-1 = 105 < 200.
Check larger magnitudes.
x = 2 gives 2·4·6·8 = 384 > 200, so any x ≥ 2 fails.
x = -8 gives -8·-6·-4·-2 = 384 > 200, so any x ≤ -8 fails.
Verify x = -3: product = -3·-1·1·3 = 9 < 200, so include -3.
Collect all integer solutions found: -7, -6, -5, -4, -3, -2, -1, 0, 1.
Count = 9 integers.
Hence the answer is "There are a total of nine values"

Question 5: Find the range of x where ||x - 3| - 4| > 3?

Answer. (-∞, -4) or (2, 4) or ( 10, ∞)

Explanation.

Set y = |x - 3| - 4. We require |y| > 3, which is equivalent to y > 3 or y < -3.

Case I: y > 3 ⇒ |x - 3| - 4 > 3 ⇒ |x - 3| > 7.

|x - 3| > 7 ⇒ x - 3 > 7 or x - 3 < -7.

Thus x > 10 or x < -4, giving intervals (-∞, -4) and (10, ∞).

Case II: y < -3 ⇒ |x - 3| - 4 < -3 ⇒ |x - 3| < 1.

|x - 3| < 1 ⇒ -1 < x - 3 < 1 ⇒ 2 < x < 4.

Combine both cases to obtain the solution set (-∞, -4) ∪ (2, 4) ∪ (10, ∞).

Hence the answer is "(-∞, -4) or (2, 4) or ( 10, ∞)"

Question 6: The sum of three distinct natural numbers is 25. What is the maximum value of their product?

Answer. 560

Explanation.
Let the numbers be a, b, c with a + b + c = 25 and a, b, c distinct natural numbers.
By AM-GM the product is maximised when the numbers are as close as possible; distinctness forces slight variation around 25/3 ≈ 8.333.
Consider integers near 8.3 while keeping them distinct.
Try (7, 8, 10): product = 7·8·10 = 560.
Try (8, 8, 9) is not allowed (not distinct).
Try nearby triples such as (6, 9, 10): product = 540; (8, 9, 8) not distinct; (5, 6, 14): product = 420.
No distinct triple with sum 25 gives product greater than 560.
Hence the answer is "The maximum product is 560."

Question 7: If x (x + 3) (x + 5) (x + 8) < 250, how many integer values can x take?

Answer. x can take integer 11 values

Explanation.
Zeros at x = 0, -3, -5, -8 give product 0, which satisfies the inequality.
Check small integers around zero.
x = 1 ⇒ 1·4·6·9 = 216 < 250, so x = 1 works.
x = 2 ⇒ 2·5·7·10 = 700 > 250, so x ≥ 2 fail.
Consider sign patterns: four factors; when the product is negative it certainly satisfies < 250. Negative product occurs when exactly one or exactly three factors are negative.
Check x = -1 and x = -2: product negative ⇒ included.
Check x = -6 and x = -7: three factors negative ⇒ product negative ⇒ included.
Check x = -9: product = -9·-6·-4·1 = 216 < 250 ⇒ included.
Check x = -10: product becomes 700 > 250 ⇒ excluded, and any smaller x will produce larger magnitude products.
Collect all integer x from -9 to 1 inclusive: -9, -8, -7, -6, -5, -3, -2, -1, 0, 1 and also -4 was checked: x = -4 ⇒ -4·-1·1·4 = 16 < 250 so include -4. Thus all integers from -9 to 1 inclusive are valid.
Count = 11 integers.
Hence the answer is "x can take integer 11 values"

Question 8: (|x| - 2) (x + 5) < 0. What is the range of values x can take?

Answer. The range is (-∞, -5) or (-2, 2)

Explanation.
For a product to be negative one factor must be negative and the other positive.
Scenario I: (|x| - 2) < 0 and (x + 5) > 0 ⇒ |x| < 2 and x > -5 ⇒ -2 < x < 2.
Scenario II: (|x| - 2) > 0 and (x + 5) < 0 ⇒ |x| > 2 and x < -5 ⇒ x < -5 (since x < -5 implies |x| > 5 > 2).
Combine the intervals: (-∞, -5) ∪ (-2, 2).
Hence the answer is "The range is (-∞, -5) or (-2, 2)"

Question 9: a and b are roots of the equation x² - p x + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?

Answer. p ≤ -8√3 or p ≥ 8√3

Explanation.
Let the roots be a and b. Then a + b = p and ab = 12.
(a - b)² = (a + b)² - 4ab = p² - 48.
The condition |a - b| ≥ 12 is equivalent to (a - b)² ≥ 144.
Therefore p² - 48 ≥ 144.
So p² ≥ 192.
Thus |p| ≥ √192 = 8√3.
Hence p ≤ -8√3 or p ≥ 8√3.
Note: real roots require discriminant p² - 48 ≥ 0, which is a weaker requirement (|p| ≥ √48) than the condition above.
Hence the answer is "p ≤ -8√3 or p ≥ 8√3"

Question 10: If a, b, c are distinct positive integers, what is the highest value a * b * c can take if a + b + c = 31?

A. 1080
B. 1200
C. 1024
D. 1056

Answer. 1080

Explanation.
For a fixed sum, the product of positive numbers is maximised when the numbers are as close as possible. Distinctness requires them to be unequal, so choose integers close to 31/3 ≈ 10.333.

Try triples near equality while keeping them distinct.
(9, 10, 12): product = 9·10·12 = 1080.
(8, 11, 12): product = 1056.
(10, 10, 11) is not allowed because numbers must be distinct.
No distinct triple with sum 31 gives a product exceeding 1080.
Hence the answer is "1080"
Choice A is the correct answer.

Question 11: a, b, c are distinct natural numbers less than 25. What is the maximum possible value of |a - b| + |b - c| - |c - a|?

A. 44
B. 46
C. 23
D. 21

Answer. 44

Explanation.
Interpret |x - y| as the distance between points x and y on the number line.
The expression |a - b| + |b - c| - |c - a| is maximised when b is far from a and c, while a and c are close together on the line and all three are distinct and within {1, 2, ..., 24}.
Choose a = 1, c = 2 and place b at the far end b = 24.
Compute the value: |a - b| + |b - c| - |c - a| = |1 - 24| + |24 - 2| - |2 - 1| = 23 + 22 - 1 = 44.
No choice can exceed this because the maximum separation available is 23 (between 1 and 24) and making a and c closer minimises the subtractive term.
Hence the answer is "44"
Choice A is the correct answer.

Question 12: Consider integers p, q such that - 3 < p < 4, - 8 < q < 7, what is the maximum possible value of p² + pq + q²?

A. 60
B. 67
C. 93
D. 84

Answer. 67
Explanation.
Possible integer values for p are -2, -1, 0, 1, 2, 3.
Possible integer values for q are -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.
Evaluate p² + pq + q² by checking extreme magnitudes, since squares dominate.
Try p = -2 and q = -7: p² + pq + q² = 4 + (-2)(-7) + 49 = 4 + 14 + 49 = 67.
Try p = 3 and q = 6: p² + pq + q² = 9 + 18 + 36 = 63.
Other combinations give smaller values because q = -7 gives the largest q² and p = -2 maximises the positive contribution of pq while keeping p² moderate.
Thus the maximum value is 67.
Hence the answer is "67"
Choice B is the correct answer.