Solve each of the following in equations and represent the solution set on the number line. |2x - 1/x-1| < 2, xϵ R.

Question

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

\(\bigg|\frac{2{\text{x}}-1}{{\text{x}}-1}\bigg|\)<2, x ϵ R.

Answer 1

Given:

Adding 2 to both sides in the above equation

Signs of 4x – 3:

4x – 3 = 0 → x = \(\cfrac{3}{4}\) 

(Adding 3 to both sides and then dividing both sides by 4)

4x – 3 > 0 → x > \(\cfrac{3}{4}\) 

(Adding 3 to both sides and then dividing both sides by 4)

4x – 3 > 0 → x < \(\cfrac{3}{4}\) 

(Adding 3 to both sides and then dividing both sides by 4) 

Signs of x – 1: 

x – 1 = 0 → x = 1 (Adding 1 to both the sides) 

x – 1 > 0 → x > 1 (Adding 1 to both the sides) 

x – 1 < 0 → x < 1 (Adding 1 to both the sides)

At x = 1 \(\frac{4{\text{x}} - 3}{{\text{x}}-1}\) is not defined.

Intervals that satisfy the required condition: > 0

Subtracting 2 from both the sides

Signs of x – 1: 

x – 1 = 0 → x = 1 (Adding 1 on both the sides) 

x – 1 < 0 → x < 1 (Adding 1 on both the sides) 

x – 1 > 0 → x > 1 (Adding 1 on both the sides)

At x = 1 \(\frac{1}{{\text{x}} - 1}\) is not defined

Interval which satisfy the required condition: < 0 

x < 1 

Now, combining the intervals:

^{x < \(\cfrac{3}{4}\)}  or x > 1 and x <1

Merging the overlapping intervals:

x < \(\cfrac{3}{4}\) 

Therefore,

x ∈ (-∞, \(\frac{3}{4}\))