Solve each of the following in equations and represent the solution set on the number line. x-7/x-2 ≥ 0, x ∈ R

Question

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

\(\cfrac{{\text{x}}-7}{{\text{x}}-2}\)\(\ge\) 0, X ∈ R

Answer 1

Given:

 \(\cfrac{{\text{x}}-7}{{\text{x}}-2}\)\(\ge\) 0, X ∈ R

 \(\cfrac{{\text{x}}-7}{{\text{x}}-2}\)\(\ge\) 0

Signs of x – 7: 

x – 7 = 0 → x = 7(Adding 7 on both the sides) 

x – 7 > 0 → x > 7 (Adding 7 on both the sides) 

x – 7 < 0 → x < 7 (Adding 7 on both the sides) 

Signs of x – 2:

x – 2 = 0 → x = 2 (Adding 2 on both the sides) 

x – 2 > 0 → x > 2 (Adding 2 on both the sides) 

x – 2 < 0 → x < 2 (Adding 2 on both the sides) 

Zeroes of denominator:

x – 2 = 0 → at x = 2 \(\cfrac{{\text{x}}-7}{{\text{x}}-2}\) will be undefined

intervals that satisfy the required condition: \(\ge\) 0

x < 2 or x = 7 or x >7

Therefore,

x є (-∞, -2) υ [7, ∞)