# Please can someone help me solve this problem?

2/x + 6/x-1 less than or = -5

8/x -1 <= -5 Now add 1 to each side

8/x <= -4 so x must be < 0

Multiply both sides by x (a negative number). This means you have to change the direction of the >

8 >= -4x

Divide both sides by 4

2 >= -x

Multiply both sides by -1

-2 <= x < 0

thanks so much but can you please post the solution in integral notation for me

2 6

___ + ____ < or = - 5

X X - 1

I'm also not sure how you got 8x-1 I didn't know I could just add the two numerators if the denominators were different

## Certainly! Let's solve the problem using integral notation.

We have the inequality:

2/x + 6/(x-1) <= -5

To solve this inequality, we'll first find the common denominator for the two fractions, which is x(x-1):

(2(x-1) + 6x) / (x(x-1)) <= -5

Simplifying the numerator:

(2x - 2 + 6x) / (x(x-1)) <= -5

(8x - 2) / (x(x-1)) <= -5

Now, we'll multiply both sides of the inequality by x(x-1) to eliminate the denominator. Since we're multiplying by a variable, we need to consider the sign:

(x(x-1))(8x - 2) / (x(x-1)) <= (-5)(x(x-1))

8x - 2 <= -5x(x-1)

Next, distribute the -5x:

8x - 2 <= -5x^2 + 5x

To solve this quadratic inequality, move all the terms to one side:

5x^2 - 3x - 2 >= 0

Now, factor the quadratic:

(5x + 2)(x - 1) >= 0

To find the solution, set each factor equal to zero and solve:

5x + 2 = 0 or x - 1 = 0

5x = -2 or x = 1

Solving these equations, we get:

x = -2/5 or x = 1

Now, we need to test the intervals using test values:

For x < -2/5, let's use x = -1:

(5(-1) + 2)(-1 - 1) >= 0

(-5 + 2)(-2) >= 0

(-3)(-2) >= 0

6 >= 0

Since this is true, x < -2/5 is part of the solution.

For -2/5 < x < 1, let's use x = 0:

(5(0) + 2)(0 - 1) >= 0

(2)(-1) >= 0

-2 >= 0

Since this is false, -2/5 < x < 1 is NOT part of the solution.

For x > 1, let's use x = 2:

(5(2) + 2)(2 - 1) >= 0

(10 + 2)(1) >= 0

12 >= 0

Since this is true, x > 1 is part of the solution.

Therefore, the solution in integral notation is:

x < -2/5 or x > 1