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in Linear Equations by (15.7k points)

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

\(\frac{1}{{\text{x}}-1} \le 2, x \in R\)

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1 Answer

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\(\frac{1}{{\text{x}}-1} \le 2, x \in R\)

Subtracting 2 from both the sides in the above equation

Signs of 3 – 2x:

3 – 2x = 0 → 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 – 2x < 0 → 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 – 2x > 0 → 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)

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) 

Zeroes of denominator: 

x – 1 = 0 → x = 1

At x = 1, \(\frac{3-2x}{x-1}\) is not defined

Intervals satisfying the condition: ≤ 0

x < 1 and x ≥ \(\frac{3}{2}\)

Therefore,

x ∈(-∞,1) U \(\bigg[ \frac{3}{2}, \infty\bigg)\)

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