Question

Solve the linear inequation, write down the solution set and represent it on the real number line:

5(2 – 4x) > 18 – 16x > 22 – 20x, x ∈ R

Answer 1

Given: 5(2 – 4x) > 18 – 16x > 22 – 20x, x ∈ R

Step-wise calculation:

1. Split the chain into two inequalities that must hold simultaneously:

1. 5(2 – 4x) > 18 – 16x 
2. 18 – 16x > 22 – 20x

2. Solve (i):

⇒ 5(2 – 4x) > 18 – 16x

⇒ 10 – 20x > 18 – 16x

⇒ –16x + 20x < 10 – 18

⇒ 4x < –8

⇒ `x < -8/4`

⇒ x < –2

3. Solve (ii):

⇒ 18 – 16x > 22 – 20x

⇒ 20x – 16x > 22 – 18

⇒ 4x > 4

⇒ `x > 4/4`

⇒ Dividing by –4 (flip inequality) 

⇒ 1 < x

⇒ x > 1

4. Combine results: x must satisfy x < –2 and x > 1 at the same time. Those intervals are disjoint, so there is no x ∈ R that satisfies both.

Solution set: ∅ the empty set.