Q5. Solve 37 − (3x + 5) ≥ 9x − 8(x − 3) for real x (NCERT Exercise 5.1, Q14).
32 − 3x ≥ x + 24, so 8 ≥ 4x and x ≤ 2: (−∞, 2].
Q6. Solve −5 ≤ (5 − 3x)/2 ≤ 8 (NCERT Example 10, page 96).
Multiply by 2: −10 ≤ 5 − 3x ≤ 16. Subtract 5: −15 ≤ −3x ≤ 11. Divide by −3 and reverse both signs: 5 ≥ x ≥ −11/3, that is −11/3 ≤ x ≤ 5.
Q7. Solve the system 2(x − 1) < x + 5 and 3(x + 2) > 2 − x (NCERT Miscellaneous Exercise, Q8).
The first gives x < 7 and the second 4x > −4, so x > −1. Together: −1 < x < 7. On the number line: hollow circles at −1 and 7, with the segment between them darkened.
Q8. Find all pairs of consecutive even positive integers, both larger than 5, whose sum is less than 23 (NCERT Exercise 5.1, Q24).
With x and x + 2 even: x > 5 and 2x + 2 < 23, so 5 < x < 10.5 and x = 6, 8 or 10: (6, 8), (8, 10) and (10, 12).
Q9. A solution must be kept strictly between 68 °F and 77 °F, with neither endpoint included. Find the range in °C, using F = (9/5)C + 32 (NCERT Miscellaneous Exercise, Q11).
68 < (9/5)C + 32 < 77 gives 36 < (9/5)C < 45, so 20 < C < 25: between 20 °C and 25 °C.
Q10. For 12-year-old children, IQ = (MA/CA) × 100 with CA = 12 and 80 ≤ IQ ≤ 140. Find the range of the mental age MA (NCERT Miscellaneous Exercise, Q14).
80 ≤ (MA/12) × 100 ≤ 140 gives 960 ≤ 100 MA ≤ 1680, so 9.6 ≤ MA ≤ 16.8 years.