Question

Solve the following equations by factorisation method.

(i) 1/x - 1/(x - 2) = 3, x ≠ 0, 2

(ii) 1/(x - 1) - 1/(x + 5) = 6/7, x ≠ 1, - 5

(iii) x - 1/x = 3, x ≠ 0

(iv) 1/(x + 4) - 1/(x - 7) = 11/30, x ≠ 4, 7

Answer 1

(i) \(\frac 1x - \frac 1{x - 2} = 3\)

⇒ 3x(x − 2) = x − 2 − x

⇒ 3x² − 6x + 2 = 0

⇒ 9x² − 18x + 6 = 0

⇒ (3x)² − 2(3x)(3) + 3² − 3 = 0

⇒ (3x − 3)² = 3

⇒ 3x − 3 = ± √3​

⇒ \(x = \frac{3 \pm \sqrt 3}3\)

⇒ \(x = 1 \pm \frac 1{\sqrt 3}\)

⇒ \(x = 1 + \frac 1{\sqrt 3}\) or \( 1 - \frac 1{\sqrt 3}\)

(ii) \(\frac 1{x - 1} - \frac 1{x + 5}= \frac 67\)

\(\frac 6{x^2 + 4x - 5} = \frac 67\)

x² + 4x - 12 = 0

x² + 6x - 2x - 12 = 0

x(x + 6) - 2(x + 6) = 0

(x + 6)(x - 2) = 0

Therefore,

x - 2 = 0

x = 2

or,

x + 6 = 0

x = -6

Hence, x = 2 or x = -6.

(iii) \( x - \frac 1x = 3\)

⇒ x² − 3x − 1 = 0

⇒ \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

⇒ a = 1, b = −3, c = −1

⇒ \(x = \frac{3 \pm \sqrt {9 + 4}}2\)

⇒ \(x=\frac{3 \pm \sqrt {13}}2\)

⇒ \(x=\frac{3 + \sqrt {13}}2\) or \(\frac{3 - \sqrt {13}}2\)

(iv) \(\frac 1{(x + 4)} - \frac 1{(x - 7)} = \frac{11}{30}\)

⇒ \(\frac{(x - 7 - x -4)}{(x + 4)(x - 7)} = \frac{11}{30}\)

⇒ \(\frac{-11}{(x + 4)(x - 7)} = \frac{11}{30}\)

⇒ −30 = x² − 3x − 28

⇒ x² − 3x + 2 = 0

⇒ x² − 2x − x + 2 = 0

⇒ x(x − 2) − 1(x − 2) = 0

⇒ (x − 2)(x − 1) = 0

⇒ x = 1, 2

Answer 2

(i) 1/x - 1/(x - 2) = 3, x ≠ 0, 2

(ii) 1/(x - 1) - 1/(x + 5) = 6/7, x ≠ 1, - 5

(iii) x - 1/x = 3, x ≠ 0

(iv) 1/(x + 4) - 1/(x - 7) = 11/30, x ≠ 4, 7