Chapter 5 Solutions — Linear Inequalities

We have 24x < 100.

Dividing both sides by 24 (positive, so inequality sign unchanged):

x < 100/24 = 25/6 ≈ 4.16

(i) When x is a natural number:

Natural numbers less than 25/6 are 1, 2, 3, 4.

Solution set = {1, 2, 3, 4}

(ii) When x is an integer:

Integers less than 25/6 are ..., −3, −2, −1, 0, 1, 2, 3, 4.

Solution set = {..., −3, −2, −1, 0, 1, 2, 3, 4}

We have 2 ≤ 3x − 4 ≤ 5.

Add 4 throughout:

2 + 4 ≤ 3x ≤ 5 + 4

6 ≤ 3x ≤ 9

Divide throughout by 3:

2 ≤ x ≤ 3

Solution set = [2, 3]

We have −12x > 30.

Dividing both sides by −12 (negative, so inequality sign reverses):

x < 30/(−12) = −5/2 = −2.5

(i) When x is a natural number:

There is no natural number less than −2.5 (all natural numbers are positive).

Solution set = {} (empty set — no solution)

(ii) When x is an integer:

Integers less than −2.5 are ..., −5, −4, −3.

Solution set = {..., −5, −4, −3}