+1678 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.)
+129895 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]
