Question 6:
List all the integer values of x which satisfy the following linear inequalities:
–2 < 3x + 1 ≤ 10

Solution:
–2 < 3x + 1
–3 < 3x
x > –1
x = 0, 1, 2, 3, …

3x + 1 ≤ 10
3x ≤ 9
x ≤ 3
x = 3, 2, 1, 0, …

Therefore x = 0, 1, 2, 3

Question 7:
List all the integer values of x which satisfy the following linear inequalities:
–5 < 2x – 3 ≤ 1

Solution:
–5 < 2x – 3
–5 + 3 < 2x
2x > –2
x > –1
x = 0, 1, 2, 3, …  

2x – 3 ≤ 1
2x ≤ 4
x ≤ 2
x = 2, 1, 0, –1, …

Therefore x = 0, 1, 2

Question 8:
$\text{Given that }3<\sqrt{x-2}<4\text{ and }x\text{ is an integer}\text{. List all the possible values of }x\text{.}$

Solution:
$\begin{array}{l}3<\sqrt{x-2}<4\\ 3^2<x-2<4^2\\ 9<x-2\\ x>11\\ \\ \text{or }x-2<16\\ x<18\\ \\ 11<x<18\\ x=12,13,14,15,16,17\end{array}$

Question 9:
Find the biggest and the smallest integer of x that satisfy
3x + 2 ≥ –4 and 4 – x > 0.

Solution:
3x + 2 ≥ –4
3x ≥ –4 – 2
3x ≥ –6
x ≥ –2

4 – x > 0
–x > –4
x < 4

Smallest integer of x is –2, and the biggest integer of x is 3.

Question 10:
$\begin{array}{l}\text{If }x\le h\le y\text{ satisfy the two inequalities }7-\frac{h}{2}\le 5\text{ and }\\ 3\left(h+2\right)\le 20+h,\text{ find the values of }x\text{ and }y.\end{array}$

Solution:
$\begin{array}{l}7-\frac{h}{2}\le 5\\ -\frac{h}{2}\le 5-7\\ -h\le -4\\ h\ge 4\\ \\ 3\left(h+2\right)\le 20+h\\ 3h+6\le 20+h\\ 2h\le 14\\ h\le 7\\ \\ 4\le h\le 7\\ \therefore x=4,y=7\end{array}$

$\begin{array}{l}\frac{5-2x}{3}\le 7\\ \left(\frac{5-2x}{3}\right)\times 3\le 7\times 3\\ 5-2x\le 21\\ -2x\le 16\\ 2x\ge -16\\ \frac{2x}{2}\ge -\frac{16}{2}\\ x\ge -8\end{array}$