Solve each of the following in equations and represent the solution set on the number line. 5x + 8/4-x < 2, x ∈ R

Question

Solve each of the following in equations and represent the solution set on the number line.

\(\frac{5{\text{x}} + 8}{4-{\text{x}}}<2, {\text{x}} \in R\)

Answer 1

Given:

\(\cfrac{5{\text{x}} + 8}{4-{\text{x}}}<2, {\text{x}} \in R\)

Subtracting both the sides by 2

\(\cfrac{5{\text{x}} + 8}{4-{\text{x}}}\) - 2< 2 - 2

Signs of x: 

x = 0 

x < 0 

x > 0 

Signs of 4 – x: 

4 – x = 0 → x = 4 

(Subtracting 4 from both the sides, then dividing by -1 on both the sides) 

4 – x < 0 → x > 4 

(Subtracting 4 from both the sides, then multiplying by -1 on both the sides) 

4 – x > 0 → x < 4 

(Subtracting 4 from both the sides, then multiplying by -1 on both the sides)

At x = 4, \(\frac{{\text{x}}}{4-{\text{x}}}\) is not defined

Intervals satisfying the condition: < 0

x < 0 or x > 4 

Therefore,

x є (-∞, 0) υ (4, ∞)