Consider the two sets: A = m R: both the roots of x^2-(m+1)x+m+4=0 are real and B = [-3,5)…

Question

Consider the two sets: A = {$m\in\mathbb R$: both the roots of $x^2-(m+1)x+m+4=0$ are real} and B = $[-3,5)$. Which of the following is not true?

Accepted Answer

$D\ge0$: $(m+1)^2-4(m+4)\ge0 \implies m^2-2m-15\ge0 \implies (m-5)(m+3)\ge0 \implies m\le-3$ or $m\ge5$.

So $A=(-\infty,-3]\cup[5,\infty)$, $B=[-3,5)$.

$$A-B = A\cap B^c = \big[(-\infty,-3]\cup[5,\infty)\big]\cap\big[(-\infty,-3)\cup[5,\infty)\big] = (-\infty,-3)\cup[5,\infty)$$

But option A states $A-B=(-\infty,-3)\cup(5,\infty)$ --- with $5$ excluded, which is wrong since $5\in A$ and $5\notin B$, so $5$ must belong to $A-B$. Hence option A is not true.

(Checking the others: $A\cap B=\{-3\}$ , $B-A=(-3,5)$ , $A\cup B=\mathbb R$  --- all correct.)

Solutions · 1