+0  
 
0
31
1
avatar+1671 

Find all real values of $x$ that satisfy $\frac{1}{x+1} + \frac{3}{x+7} \ge \frac{1}{2}.$ (Give your answer in interval notation.)

 Sep 9, 2023
 #1
avatar+128397 
+1

1/ ( x + 1) + 3 / (x +7) ≥  1/2

 

[ (x + 7) +  3(x + 1) ] / [(x + 1)(x+ 7)] - 1/2 ≥  0

 

[ 4x + 10 ]  / [ ( x + 1 ) (x + 7) ]   - 1/2 ≥  0

 

[ 2 (4x + 10) - (x + 1) ( x + 7) ] / [ 2 (x + 1) (x + 7) ] ≥  0

 

[ 8x + 20 - x^2 - 8x - 7 ] / [ 2 ( x + 1) (x + 7) ] ≥  0

 

[ -x^2 + 13 ] / [ (x + 1) ( x + 7) ]  ≥  0          (1)

 

We  have  five  possible intervals to  check

(-inf , -7)          check with x = -8   ....this interval makes (1) false

(-7 , -sqrt 13]  check with x =  -5   ....this interval makes (1) true

[-sqrt 13 , -1)  check with x = -2  ......this interval makes (1) false

(-1 , sqrt 13 ]  check with x = 0   ......this interval makes (1) true

[sqrt 13 , inf)  check with x = 5   ......this interval  makes (1) false

 

Solution comes from  

 

( -7 , -sqrt 13]  U  ( -1 , sqrt 13]

 

cool cool cool

 Sep 10, 2023

3 Online Users

avatar
avatar