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