Question 61887: Couls someone help please?
Solve the following inequalitites. Write the answers in interval notation.
x^2 +7x-18=>0
3x-4/2x+1 <=6
Answer by jai_kos(139)   (Show Source):
You can put this solution on YOUR website!
Step 1: Write the quadratic inequality in standard form.
This quadratic inequality is already in standard form
Step 2: Solve the quadratic equation,, by factoring to get the boundary point(s).
x^2 +7x-18 =>0
x^2 + 9x -2x -18 = >0
x(x + 9) - 2 (x - 9) => 0
(x -2) (x-9) = > 0
(x -2) = 0 or (x -9) = 0 ---->(1)
x =2 or x = 9
2 and 9 are boundary points.
Step 3:Considering the boudary points, we construct the intervals.
The intervals are (- infinty,2) (2,9) and (9,+infinity)
Step 4: Find the sign of every factor in every interval.
Consider the first interval ( -infinty ,2)
Take a point in the interval, let it be 1.
Now put in equation(1) we get (x - 2) = 1 -2 = -1
(x -9) = 1 -9 = -8
Both the values are negative.
Consider the second interval (2,9),
Take a point in the interval, let it be 3.
Now put in equation(1), we get
(x - 2) = 3 -2 = 1
(x -9) = 3 -9 = -6
One value is positive and the other song is negative.
Consider the Third interval (9, infinity),
Take a point in the interval, let it be 10.
Now put in equation(1), we get
(x - 2) = 10 -2 = 8
(x -9) = 10 -9 = 1
Both the values are positive.
So we consider the interval in which there is variation.
That is,
(2 ,9)
The quadratic equation in the interval notation, we have (2, 9)
2) 3x-4/2x+1 <=6
Mulitply the above equation by (2x +1), we get
3x - 4< = 6 (2x +1)
3x -4 < = 12x + 6
Group all the "x" terms on one side, we get
3x + 12x < = 4 + 6
15x < = 10
Dvide the above equation by 15, we get
x< =10 /15
x <=2/3
Therefore the interval would be (-infinity ,2/3]
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