Find the solution set for each rational inequation. Graph the solution set on a number line.

Question

1.)(X+3)(x-2)/(x+2)(x-1) greater than or equal to 0. 2.) (X+4)(x-3)/(x-2)(x^2+2) greater than or equal to 0. 3.)X+1/x+3 less than or equal to 2. 4.)X-2/x^2-3x-10 less than 0. Email me if this is confusing, thanks.

Answer 1

1. (x+3)(x-2)/((x+2)(x-1))≥0. We need to look at various intervals marked by -3, -2, 1, 2 on the number line. x≤-3 satisfies the inequality; -3<x<-2 fails the inequality; x=-2 and x=1 are disallowed because we would be dividing by 0; -2<x<1 satisfies the inequality; 1<x<2 fails; x≥2 satisfies the inequality.
2. (x+4)(x-3)/((x-2)(x^2+2))≥0. Significant values are -4, 2, 3; x=2 is disallowed; x<-4 fails; -4≤x<2 satisfies; 2<x<3 fails; x≥3 satisfies.
3. (x+1)/(x+3)≤2, (x=-3 is disallowed) x+1≤2x+6, -5≤x, x≥-5 satisfies the inequality, but -3 is disallowed (hole). Therefore -5≤x<-3 and x>-3.
4. (x-2)/(x^2-3x-10)<0. This is (x-2)/((x-5)(x+2))<0. Disallowed values are 5 and -2. Significant values are -2, 2, 5; x<-2 satisfies; -2<x≤2 fails; 2<x<5 satisfies; x>5 fails.