Question

What is the solution of (2/x-1)>(7/x+1)

Answer 1

(2 / x-1)> (7 / x + 1)
 For this case we can solve the problem graphically.
 The solution will be the shaded region.
 Note: see attached image.
 Answer:
 The solution is given by:1 <x <9/5

Answer 2

Answer:

Given inequality,

$\frac{2}{x} - 1 > \frac{7}{x} + 1$

Using the property a > b ⇒ a + c > b + c ∀ a, b, c ∈ R

$\frac{2}{x} > \frac{7}{x}+2$

Using the property a > b ⇒ a - c > b - c ∀ a, b, c ∈ R

$\frac{2}{x}-\frac{7}{x} > 2$

$\frac{2-7}{x} > 2$

$-\frac{5}{x} > 2$

Case 1 : If x > 0 ⇒ -x < 0

∵ a > b ⇒ ca < cb if c < 0,

$\implies 5 < -2x$

Now, -1/2 < 0

$\implies -\frac{5}{2} > x$

Case 2 : If x < 0 ⇒ -x > 0

∵ a > b ⇒ ca > cb if c > 0,

$\implies 5 > -2x$

Now, -1/2 < 0

$\implies -\frac{5}{2} < x$