4/x^2 - 4x + 3 < 5/x^2 - 9
This is a rational inequality. it is 4 divided by x squared minus 4x plus 3 is less than or equal to 5 divided by x squared minus 9. Please help!!!
Please Help ME!!!!!!!!!!!!!!!!!
Get things organized a bit:
1/x^2 + 4x - 12 > 0
Now, as long as x is not zero, we can multiply by x^2
1 + 4x^3 - 12x^2 > 0
The roots of this cubic are
-0.276, 0.305, 2.972
y>0 on (-0.276,0) (0,0.305) (2.972,+oo)
To solve this rational inequality, we'll need to follow these steps:
Step 1: Find the common denominator
Multiply both sides of the inequality by the common denominator of the fractions involved. In this case, the common denominator is (x^2 - 9) since it appears in both fractions.
(x^2 - 9) * (4/x^2 - 4x + 3) < (x^2 - 9) * (5/x^2 - 9)
Simplifying this expression yields:
4(x^2 - 9) < 5(x^2 - 4x + 3)
Step 2: Expand the terms
Distribute on both sides of the inequality to expand the expressions.
4x^2 - 36 < 5x^2 - 20x + 15
Subtract 4x^2 from both sides to move all terms to one side:
-36 < x^2 - 20x + 15
Step 3: Set the inequality equal to zero
Since it is a rational inequality, we need to set it equal to zero.
0 < x^2 - 20x + 15 + 36
0 < x^2 - 20x + 51
Step 4: Solve the quadratic equation
To solve the quadratic equation, we can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -20, and c = 51.
x = (-(-20) ± √((-20)^2 - 4(1)(51))) / (2(1))
x = (20 ± √(400 - 204)) / 2
x = (20 ± √196) / 2
x = (20 ± 14) / 2
Simplifying further:
x = (20 + 14) / 2 = 34 / 2 = 17
x = (20 - 14) / 2 = 6 / 2 = 3
Step 5: Determine the intervals of the solution
Now that we have the solutions, we can determine the intervals where the original inequality holds true. To do this, we create a number line and test values in each interval.
The intervals are (-∞, 3) and (3, 17).
Take a value from each interval and substitute it into the original inequality (e.g., x = 0 for the interval (-∞, 3), and x = 10 for the interval (3, 17)). If the inequality holds true for those values, then the interval is part of the solution. If it doesn't, then it is not part of the solution.
For interval (-∞, 3):
4/(0)^2 - 4(0) + 3 < 5/(0)^2 - 9
Since dividing by 0 is undefined, we exclude this interval.
For interval (3, 17):
4/(10)^2 - 4(10) + 3 < 5/(10)^2 - 9
Simplifying, we get:
-3.9 < -9.45
Since this inequality is false, we exclude the interval (3, 17) from the solution.
Step 6: Final solution
The solution to the original inequality 4/x^2 - 4x + 3 < 5/x^2 - 9 is the interval (-∞, 3) excluding the value 3.
Therefore, the answer is x < 3.