**Question:**

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

$\frac{1}{x-1} \leq 2, x \in R$

**Solution:**

Given:

$\frac{1}{x-1} \leq 2, x \in R$

Subtracting 2 from both the sides in the above equation

$\frac{1}{x-1}-2 \leq 2-2$

$\frac{1-2(x-1)}{x-1} \leq 0$

$\frac{1-2 x+2}{x-1} \leq 0$

$\frac{3-2 x}{x-1} \leq 0$

Signs of 3 – 2x:

$3-2 x=0 \rightarrow x=\frac{3}{2}$

(Subtracting by 3 on both the sides, then multiplying by $-1$ on both the sides and then dividing both the sides by 2 )

$3-2 x<0 \rightarrow x>\frac{3}{2}$

(Subtracting by 3 on both the sides, then multiplying by $-1$ on both the sides and then dividing both the sides by 2)

$3-2 x>0 \rightarrow x<\frac{3}{2}$

(Subtracting by 3 on both the sides, then multiplying by $-1$ on both the sides and then dividing both the sides by 2 )