Question

Please answer the inequality as fast as you can

Answer 1

Answer:

x<6. and x< 1

Step-by-step explanation:

14x/x+1< 9x-30/x-4

14x(x-4) < (x+1)(9x-30)

14x^2-56x < 9x^2-21x-30

14x^2-9x^2-56x+21x<-30

5x^2- 35x <-30

dividing both sides by 5

x^2-7x<-6

x^2 -7x+6<0

x^2 -6x-1x+6<o

x(x-6) -1(x-6)<0

(x-6)(x-1)<0

x-6<0. or. x-1<0

x<6. or. x<1

Answer 2

✪ANSWER✪

• $\boxed{x ∈ (-1, 1)∪(4, 6)}$

★EXPLAINATION★

The given inequality is

$\frac{14x}{x + 1} < \frac{9x - 30}{x - 4}$

$\frac{14x}{x + 1} - \frac{9x - 30}{x - 4} < 0$

$\frac{14x(x - 4) - (x + 1)(9x - 30)}{(x + 1)(x - 4)} < 0$

$\frac{(14 {x}^{2} - 56x) - (9 {x}^{2} - 21x - 30)}{(x + 1)(x - 4)} < 0$

$\frac{5 {x}^{2} - 35x + 30}{(x + 1)(x - 4)} < 0$

$\frac{ {x}^{2} - 7x + 6 }{(x + 1)(x - 4)} < 0$

$\frac{ {x}^{2} - x - 6x + 6 }{(x + 1)(x - 4)} < 0$

$\frac{x(x - 1) - 6(x - 1)}{(x + 1)(x - 4)} < 0$

$\frac{(x - 1)(x - 6)}{(x + 1)(x - 4)} < 0$

᯽SIGN OF THE EXPRESSION᯽

Now, let us draw a number line and mark the points -1, 1, 4, 6 on it. It divides the number line into 5 section. We will check the sign of the above expression on each section.

1) x < -1

In this case all factors i.e. (x-1),(x-6), (x+1), (x-4) are negative. Hence sign of the expression is positive. Thus this section is not a solution of the inequality

2) -1 < x < 1

In this case (x-1) is positive and (x-6), (x+1), (x-4) are negative. Hence sign of the expression is negative. Thus this section is a solution of the inequality.

3) 1 < x < 4

In this case, (x-1), (x+1) are positive and (x-4), (x-6) are negative. Hence sign of the expression is positive. Thus this section is not a solution of the inequality

4) 4 < x < 6

In this case (x-6) is negative and (x-1), (x+1), (x-4) are positive. Hence sign of the expression is negative. Thus this section is a solution of the inequality.

5) x > 6

In this case, all factors i.e. (x-1),(x-6), (x+1), (x-4) are positive. Hence sign of the expression is positive. Thus this section is not a solution of the inequality.

From the above analysis, the expression has negative values when

$-1 < x < 1\:\: or\:\: 4 < x < 6$

$⇒ x ∈ (-1, 1)\:\:or\:\: x ∈(4, 6)$

$⇒ x ∈ (-1, 1)∪(4, 6)$