x^2 +2x -3>0
I don't understand when to use the union symbol After using my three test points. i know the answer is (x-1)(x+3) so x = -3 X= 1
You are asking for a hard and fast rule which does not exist.
In general, quadric inequalities can have the whole line as a solution set, such as $\displaystyle x^2+1\ge 0~.$
Or the solution set can be between to real numbers, as in $\displaystyle (x+2)(x-3)\le 0$ the set is $\displaystyle [-2,3]$.
Out side two numbers: $\displaystyle (x-2)(x+3)\ge 0$, solution $\displaystyle (-\infty ,-3]\cup [2,\infty)$.
But that is only a rule of thumb.
Let $\displaystyle \[f\left( x \right) = a{x^2} + bx + c\]$
- If D>0, then f(x) has two real roots. Between those the sign of f(x) is the opposite of the sign of $\displaystyle \[a\]$ and outside them it has the same sign
- If D=0, f(x) has the same sign with a except the point of zero.
- If D<0, f(x) has the same sign with a
Are you saying that you do not know what the union symbol means?
The fact that x= -3 and x= 1 make the two sides equal tells you that the inequality is one way or the other through out the three intervals, $\displaystyle (-\infty, -3)$, $\displaystyle (-3, 1)$, and $\displaystyle (1, \infty)$. Plato told you that the solution cannot be (1, 3) because x= 0 is in that interval and 0 does not satisfy the inequality. You really should, then, check one point in each of the other intervals. x= 2 is in the interval $\displaystyle (1, \infty)$. $\displaystyle 2^2+ 2(2)- 3= 4+ 4- 3= 5> 0$ so x= 2 and, so, every number in $\displaystyle (1, \infty)$ satisfies it. x= -4 is in $\displaystyle (-\infty, -3)$ and $\displaystyle (-4)^2+ 2(-4)- 3= 16- 8- 3= 5> 0$ so x= -4 and, therefore, every number in $\displaystyle (-\infty, -3)$ satisifes the inequality.
That is, any number in $\displaystyle (-\infty, -3)$ or in $\displaystyle (1, \infty)$ will satisfy the inequality. A point is in $\displaystyle A\cup B$ if and only if it is in A or in B, by definition.
You answered the question. Thank you, for understanding what I was trying to convey. I assumed I broke it clearly down to what I did and didn't understand, but I guess people are in a rush and only wish to explain the answer. Thank you so much for being patient and throughly explaining everything.