12.2.1 Linear Inequalities, PT3 Focus Practice

12.2.1 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.

Solution:
3x – 5 ≤ 1
3x  ≤ 1 + 5
3x  ≤ 6
x  ≤ 2

Question 3:
The solution for the inequality 2 + < 3x – 4 is

Solution:
2 + x < 3x – 4
x – 3x < –4 – 2
–2x < –6
x < –3
x > 3

Question 4:
The solution for the inequality –2 (6y + 3) < 3 (4 – 2y) is

Solution:
–2 (6y + 3) < 3 (4 – 2y)
–12y – 6 < 12 – 6y
–12y + 6< 12 + 6
–6y < 18
y < 3
y > –3

Question 5:
Solve each of the following inequalities.
(a) 3x + 4 > 5x – 10
(b)   –3  ≤ 2x + 1 < 7

Solution:
(a)
3x + 4 > 5x – 10
3x – 5x > –10 – 4
–2x > –14
x > –7
x < 7

(b)
–3 ≤ 2x + 1 < 7
–3 ≤ 2x + 1   and   2x + 1 < 7
–2x  ≤ 1 + 3   and   2x < 7 – 1
–2x  ≤ 4 and   2x < 6
≥ –2   and   x < 3

The solution is –2 ≤ x < 3.