For what values of $x$ is $$\frac{x^2 + x + 3}{2x^2 + x - 6} \ge 0? $$ note: be thorough and explain why all points in your answer are solutions and why all points outside your answer are not solutions.

To solve the inequality

The numerator is not factorizable.

so factor the denominator :

Now take

Then we get factor of the denominator as

Thus

Now

Signs of

is the required solution of the given inequality.

Step-by-step explanation:

Here, the given expression,

Since, for the value of x,

Denominator ≠ 0,

Thus, there is three intervals possible,

1)

2)

3)

In first intervals, is  true.

⇒ will contain the value of x.

In second interval,

is not true,

⇒ will not contain the value of x,

In third interval,

In third interval,

is true,

⇒ will contain the value of x,

In third interval,

Thus, the value of x is,

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation: