Question 61887

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]