"]If x is an integer, what is the value of x?
(1) x2 - 4x + 3 < 0
(2) x2 + 4x +3 > 0
(1) \(x^2-4x+3<0\) --> \(1<x<3
\)--> as \(x\) is an integer then \(x=2\). Sufficient.
(2) \(x^2+4x+3>0\) --> \(x<-3\) or \(x>-1\) --> multiple values are possible for integer \(x\). Not sufficient.
With reference to your statement \(1<x<3\) above, I calculated the range as \(x<1\) or \(x<3\)
What I did wrong?[/quote]How to solve quadratic inequalities - Graphic approach.
\(x^2-4x+3<0\) is the graph of parabola and it look likes this:
Intersection points are the roots of the equation \(x^2-4x+3=0\), which are \(x_1=1\) and \(x_2=3\). "<"
sign means in which range of \(x\) the graph is below
x-axis. Answer is \(1<x<3\) (between the roots).If the sign were ">"
: \(x^2-4x+3>0\). First find the roots (\(x_1=1\) and \(x_2=3\)). ">"
sign means in which range of \(x\) the graph is above
x-axis. Answer is \(x<1\) and \(x>3\) (to the left of the smaller root and to the right of the bigger root).
This approach works for any quadratic inequality. For example: \(-x^2-x+12>0\), first rewrite this as \(x^2+x-12<0\) (so that the coefficient of x^2 to be positive. It's possible to solve without rewriting, but easier to master one specific pattern).
\(x^2+x-12<0\). Roots are \(x_1=-4\) and \(x_1=3\) --> below ("<"
) the x-axis is the range for \(-4<x<3\) (between the roots).
Again if it were \(x^2+x-12>0\), then the answer would be \(x<-4\) and \(x>3\) (to the left of the smaller root and to the right of the bigger root).
Hope it helps.[/quote]
I am fond of your explanations !!