Practicing Success
The set of all values of x, measured in radians, satisfying the two inequalities $2 \cos^2 x <2-3 \cos x$ and $x^2 <4x+12$, is |
$(-2, -π/3)∪(π/3, 5π/3)$ $(π/3, π/2)∪(3π/2, 2/π)$ $(-2,6)$ $(-2,1/2)$ |
$(-2, -π/3)∪(π/3, 5π/3)$ |
We have, $2 \cos^2 x <2-3 \cos x$ and $x^2 <4x+12$ Now, $2 \cos^2 x <2-3 \cos x$ $⇒2 \cos^2 x + 3 \cos x-2<0$ $⇒(2 \cos x-1) (\cos x + 2) <0$ $⇒2 \cos x-1 <0⇒ \cos x<\frac{1}{2}$ $[∵\cos x+2>0]$ ...(i) and, $x^2 <4x + 12$ $⇒x^2-4x-12<0⇒(x-6) (x+2) <0⇒>-2<x<6$ ...(ii) Now, we have to find the values of x which satisfy in equations (i) and (ii). We know that $-π <-2<-\frac{π}{2}<\frac{π}{2}<2<π<\frac{3π}{2}<6<2π$ $∴\cos x<\frac{1}{2}$ if $x∈(-2, -π/3)∪(π/3, 5π/3)$ Hence, the set of values of x satisfying the given inequations is $(-2, -π/3)∪(π/3, 5π/3)$ |