Q3
Find all pairs of consecutive even positive integers, both of which are larger than 5, such that their sum is strictly less than 23.
(6, 8), (8, 10)
(8, 10), (10, 12)
(6, 8), (8, 10), (10, 12)
(4, 6), (6, 8)
x > 5 and x + x + 2 < 23 → 2x < 21 → x < 10.5. Valid even x: 6, 8, 10. The pairs are (6, 8), (8, 10), and (10, 12).
Q5
The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter is at least 61 cm, find the minimum length of the shortest side.
7 cm
8 cm
9 cm
10 cm
Let shortest side be x. Longest = 3x. Third = 3x − 2. x + 3x + 3x − 2 ≥ 61 → 7x − 2 ≥ 61 → 7x ≥ 63 → x ≥ 9.
Q7
Solve (2x − 1)/(x + 1) ≤ 1.
(−1, 2]
[−1, 2]
(−∞, −1) ∪ [2, ∞)
(−1, 2)
(2x − 1)/(x + 1) − 1 ≤ 0 → (2x − 1 − x − 1)/(x + 1) ≤ 0 → (x − 2)/(x + 1) ≤ 0. Critical points −1, 2. x ∈ (−1, 2].
Q8
A train journey takes less than 12 hours. If t is time in hours, then:
t < 12
t ≤ 12
t > 12
t ≥ 12
Less than indicates strict inequality.
Q10
The solution set of all real numbers x satisfying x < 1 or x > 4 is:
(−∞, 1) ∪ (4, ∞)
(1, 4)
[1, 4]
(−∞, 4)
Values less than 1 or greater than 4 form two separate intervals.
Q12
Find x if 0.5x − 2.5 ≥ 1.5x − 0.5.
x ≤ −2
x ≥ −2
x ≤ 2
x ≥ 2
Subtract 0.5x: −2.5 ≥ x − 0.5. Add 0.5: −2.0 ≥ x, or x ≤ −2.
Q13
Identify the inequality whose solution is the set of all real numbers x such that distance from x to 5 is less than 3.
|x − 5| < 3
|x + 5| < 3
|x − 3| < 5
|x + 3| < 5
Distance between x and 5 is |x − 5|. So, |x − 5| < 3.
Q14
A warehouse can store no more than 800 boxes. If x is the number stored, then:
x ≤ 800
x ≥ 800
x < 800
x > 800
No more than means less than or equal to.
Q15
If |x + 2| ≤ 9 and x is an integer, how many solutions exist?
17
18
19
20
−9 ≤ x + 2 ≤ 9 → −11 ≤ x ≤ 7. Number of integers = 7 − (−11) + 1 = 19.