**12.2 Linear Inequalities, PT3 Focus Practice**

**Question 1:**

Draw a number line to represent the solution for the linear inequalities –3 < 5 –

*x*≤ 4.

*Solution:*–3 < 5 –

*x*and*5 –**x*≤ 4*x*< 5 + 3

*and*

*–*

*x*≤ 4 – 5

*x*< 8

*and*

*–*

*x*≤ –1 →

*x*≥ 1

Thus, the solution is 1 ≤

*x*< 8

**Question 2:**

Solve the following simultaneous linear inequalities.

$3x-5\le 1\text{and}2-\frac{1}{3}x<3$

Solution:Solution:

3

*x*– 5 ≤ 13

*x*≤ 1 + 53

*x*≤ 6*x*≤ 2

$\begin{array}{l}2-\frac{1}{3}x<3\\ 6-x<9\leftarrow \overline{)\text{Multiply by 3}}\\ -x<9-6\\ -x<3\\ x>-3\leftarrow \overline{)\text{Multiply by}-1}\end{array}$

The solution is

**–3 <**.*x*≤ 2**Question 3:**

The solution for the inequality 2 +

*x*< 3*x*– 4 is

*Solution:*2 +

*x*< 3*x*– 4*x*– 3

*x*< –4 – 2

–2

*x*< –6–

*x*< –3

*x***> 3**

**Question 4:**

The solution for the inequality –2(6

*y*+ 3) < 3(4 – 2*y*) is

*Solution:*–2(6

*y*+ 3) < 3(4 – 2*y*)–12

*y*– 6 < 12 – 6*y*–12

*y*+ 6*y*< 12 + 6–6

*y*< 18–

*y*< 3

*y***> –3**

**Question 5:**

Solve each of the following inequalities.

(a) 3

*x*+ 4 > 5*x*– 10(b) –3 ≤ 2

*x*+ 1 < 7

*Solution:*(a)

3

*x*+ 4 > 5*x*– 103

*x*– 5*x*> –10 – 4–2

*x*> –14–

*x*> –7

*x***< 7**

(b)

–3 ≤ 2

*x*+ 1 < 7–3 ≤ 2

*x*+ 1 and 2*x*+ 1 < 7–2

*x*≤ 1 + 3 and 2*x*< 7 – 1–2

*x*≤ 4 and 2*x*< 6*x*≥ –2 and

*x*< 3

The solution is

**–2 ≤**.*x*< 3