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  • Solve the following system of inequalities 2x plus 1 divided by 7x minus 1 greater than 5 x plus 7 divide- by x minus 8 greater than 2

Solve the following system of inequalities: \frac{2x+1}{7x-1}> 5,\frac{x+7}{x-8}>2

Answers (1)

Given: 2x+1/7x-1 > 5    

                (2x+1)/(7x - 1) – 5 > 0     …….. (on subtracting 5 from both sides)

            (2x + 1 – 35x + 5)/(7x – 1) > 0

            (6 – 33x)/(7x – 1) >0

Now, either the numerator or the denominator should be greater than 0 or both should be less than 0 for the above fraction to be greater than 0. Thus,

6 – 33x > 0 & 7x – 1 > 0

33x <6        & 7x > 1

X < 2/11     & x > 1/7

i.e., 1/7 < x < 2/11     ……. (i)

Or,

6 – 33x < 0 & 7x – 1 < 0

33x > 6       & 7x < 1

X > 2/11     & x < 1/7

i.e., 2/11 < x < 1/7   ….. viz. impossible

Now,

(x + 7)/(x – 8) > 2    …… (given)

(x + 7)/(x – 8 )– 2 > 0       …… (on subtracting both sides by 2)

(x +7 – 2x + 16)/(x – 8) > 0

(23 – x)/(x – 8) > 0

Now, either the numerator or the denominator should be greater than 0 or both should be less than 0 for the above fraction to be greater than 0. Thus,

23 – x > 0 & x – 8 > 0

X < 23       & x > 8

i.e., 8 < x < 23           …….. (ii)

Or,

23 – x > 0  & 8 > 0

X > 23        & x < 8

i.e., 23 < x < 8       ….. viz. impossible

Therefore, from (i) & (ii) we can say that there is no solution satisfying both inequalities.

Thus, the system has no solution.
 

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