Page 174 - ICSE Math 8

Solution: 3x + 2 ≤ 8
⇒ 3x + 2 – 2 ≤ 8 – 2 ⇒ 3x ≤ 6 (Subtracting 2 from both sides)
6
⇒ 3x ≤ ⇒ x ≤ 2 (Dividing both sides by 3)
3 3
But x ∈ {1, 2, 3, 4, 5, 6, 7}, so the possible values of x are {1, 2}.
To represent this solution set graphically, make thick dots on numbers 1 and 2.

1 2 3 4 5 6 7
Example 18: Solve –x + 1 < 3, x ∈ R. Represent the solution set on a number line.
–x 3
Solution: 3 + 1 < 3
⇒ –x + 1 – 1 < 3 – 1 ⇒ –x < 2 (Subtracting 1 from both sides)
3
3
⇒ –3 × –x > 2 × (–3) ⇒ x > – 6 (Multiplying both sides by –3)
3
To represent this solution set on a number line, draw a hollow circle over –6 and draw a dark
line with dark arrow on the right of –6 which shows that every number greater than –6 is also
included in the solution.

–7 –6 –5 –4 –3 –2 –1 0 1 2 3
Example 19: Solve 15x ≥ 3(x – 2), x ∈ W and represent it graphically.
Solution: 15x ≥ 3(x – 2) Try This
⇒ 15x ≥ 3x – 6 ⇒ 15x – 3x ≥ –6 (Transposing 3x) Find the solution set of
12x –6 13 > 2x – 5 > 5, x ∈ R and
⇒ 12x ≥ –6 ⇒ 12 ≥ 12 (Dividing both sides by 12)
–1 represent it graphically.
⇒ x ≥
2
But x ∈ W, so x = {0, 1, 2, ...}.
∴ The solution set = {0, 1, 2, ...}
To represent this solution set on number line, mark 0, 1, 2, ... by thick dots.
... →
or
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
Points to remember

• The arrow or three dots above the number line show that the subsequent whole numbers are also in the
solution set.
• A solid circle is used on a number line when that value is included in the solution set.
• A hollow circle is used on a number line when that value is not included in the solution set.

EXERCISE 15.5

1. Find the solution set for the following inequations when the replacement set is
{0, 1, 2, 3, 4, 5, 6, 7, 8}.
(a) x ≥ 5 (b) x ≤ 4 (c) x + 2 ≥ 8 (d) 5 – x ≤ 3
3x – 1
(e) 7x < 8 (f) 15 – 3x ≤ 9 (g) 3(x – 2) ≤ 2x – 1 (h) 2x – 5 <
2
2. Represent the solution set graphically.
(a) x – 3 ≤ –10, x ∈ R (b) –63 + x ≥ –70, x ∈ {Set of negative integers}
(c) x – 15 ≥ –13, x ∈ {–3, –2, –1, 0, 1, 2, 3} (d) 1 x < 2, x ∈ Z
3
(e) 17 + x > 20, x ∈ W (f) x + 4 ≥ 9, x ∈ N

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