2 6
___ + ____ < or = - 5
X X - 1
Can some one please clearly show me how to answer this and put the solution in interval notation?
You need to find another way to write the inequality more clearly. Try using / for division and parentheses for clarification of operations. Do you mean
(2/x) + [6/(x-1)] <or= -5 ?
Try rewriting with a common denominator for all terms first. That will help you get rid of the reciprocals 1/x and 1/(x-1) . Do the = case first and worry about the inequality case later.
To rewrite the equation with a common denominator, we multiply each fraction by the appropriate factor to eliminate the denominators.
In this case, the common denominator is x(x-1).
So, we can rewrite the equation as:
(2(x-1) + 6x) / (x(x-1)) <= -5
Now, let's simplify the equation further:
(2x - 2 + 6x) / (x(x-1)) <= -5
Combining like terms:
(8x - 2) / (x(x-1)) <= -5
Next, let's solve for the equal case:
(8x - 2) / (x(x-1)) = -5
To get rid of the denominator, we can multiply both sides of the equation by x(x-1):
(x(x-1)) * (8x - 2) / (x(x-1)) = -5 * x(x-1)
Simplifying:
8x - 2 = -5x^2 + 5x
Rearranging and combining like terms:
5x^2 - 3x - 2 = 0
Now, solving this quadratic equation can be done using factoring, completing the square, or the quadratic formula. For the sake of brevity, I'll use the quadratic formula:
x = (-b +/- sqrt(b^2 - 4ac)) / (2a)
Plugging in the values from our quadratic equation:
x = (-(-3) +/- sqrt((-3)^2 - 4*5*(-2)) / (2*5)
Simplifying:
x = (3 +/- sqrt(9 + 40)) / 10
x = (3 +/- sqrt(49)) / 10
x = (3 +/- 7) / 10
This gives us two possible solutions:
x = (3 + 7) / 10 = 10 / 10 = 1
x = (3 - 7) / 10 = -4 / 10 = -2/5
Now, we can move on to the inequality case.
(8x - 2) / (x(x-1)) <= -5
First, we need to find the critical points where the expression is undefined. These are the values that make the denominator zero, in this case, x = 0 and x = 1.
We can now create a number line and test different intervals to find out where the expression is less than or equal to -5.
Interval 1: (-∞, 0)
We can choose a test point, let's say -1, to check the expression.
(8(-1) - 2) / ((-1)(-1 - 1)) = (-10) / (2) = -5
Since the expression is equal to -5, it satisfies the condition.
Interval 2: (0, 1)
Similarly, we can choose a test point, let's say 0.5, to check the expression.
(8(0.5) - 2) / ((0.5)(0.5 - 1)) = 5
Since 5 is not less than or equal to -5, it does not satisfy the condition.
Interval 3: (1, ∞)
Again, we can choose a test point, let's say 2, to check the expression.
(8(2) - 2) / ((2)(2 - 1)) = 14 / 2 = 7
Since 7 is not less than or equal to -5, it does not satisfy the condition.
Therefore, the solution to the inequality is x = (-∞, 0] U (1, ∞) in interval notation.