Please help with the second part of this question Please exp

Please help with the second part of this question.. Please explain how too... Thank you

Solve the rational inequality indicated using a number line and the behavior of the graph at each zero. v(x) = x^2 + 4x/x^2 - x - 12; v (x) > 0 The factored form is v(x) = x(x + 4)/(x - 4)(x + 3). The solution in interval notation is x

Solution

factored form is

x (x + 4) / ( x - 4) ( x + 3)

to find critical points

set numerator and denominator equal to 0 and solve for x

x ( x+ 4) = 0

x = 0 , x = -4

( x-4) (x+3) = 0

x = 4 , x = - 3

critical points are 0 , - 4 , 4 , -3

dividing the number line into 5 intervals

------------- -4 --------------- - 3 ----------------- 0 ---------------------- 4 -----------------

1st interval ( - inf to -4) test point -5

x (x + 4) / ( x - 4) ( x + 3) > 0

-5 ( -5 + 4) / (-5-4)(-5+3)

positive / positive > 0

so its true

2nd interval ( -4 to -3 ) , test point -3.5

-3.5 (-3.5 + 4) / ( -3.5 - 4) ( -3.5 + 3) >0

negative / positive = negative which is less than 0

hence , false

3rd interval ( -3 to 0 ) , test point -1

-1 (-1 + 4) / ( -1 - 4) ( -1 + 3)

negative / negative = positive which is greater than 0

hence, true

4th interval ( 0 , 4 ) , test point 1

1 (1 + 4) / ( 1 - 4) ( 1 + 3)

positive / negative = negative which is less than 0

hence, false

5th interval ( 4 , inf ) test point = 5

5 (5 + 4) / ( 5 - 4) ( 5 + 3)

positive / positive

true

solution in interval notation is

( - infinity to - 4 ) U ( - 3 to 0 ) U ( 4 to infinity )