Solved Example
Question 1
$x^2-5x+6 > 0 $
Solution
1) Simpify or factorize the inequality which means factorizing the equation in case of quadratic equalities
Which can be simpified as
$x^2-5x+6 > 0 $
$x^2 -2x-3x+6 > 0$
$(x-2)(x-3) > 0 $
2) Now plot those points on Number line clearly
3) Now start from left of most left point on the Number line and look out the if inequalities looks good or not. Check for greater ,less than and equalities at all the end points
So in above case of
$x^2-5x+6 > 0$
We have two ends points 2 ,3
Case 1
So for $x < 2$ ,Let take $x=1$,then $(1-2)(1-3) > 0$
$2 > 0$
So it is good
So This inequalties is good for x < 2
Case 2
Now for x =2,it makes it zero,so not true. Now takes the case of $2< x < 3$. Lets takes 2.5
$(2.5-2)(2.5-3) > 0$
$-.25 > 0$
Which is not true so this solution is not good
Case 3
Now lets take the right most part i.e $x > 3 $
Lets take x=4
$(4-2)(4-3) > 0$
$2> 0$
So it is good.
Now the solution can either be represented on number line or we can say like this
$(-\infty,2)\cup (3,\infty)$
Practice Questions
- $x^2 -7x +10 > 0 $
- $x^2 -2x -15 < 0 $
- $x^2 +2x +1 > 0 $
- $x^{2} - 13 x + 36 > 0$
- $x^{2} - 8 x + 7 < 0$
- $x^{2} - 12 x + 20 \geq 0$
- $x^{2} - 11 x + 28 \leq 0$
- $x^{2} - 6 x + 8 \geq 0$
- $x^{2} - 8 x + 12 \leq 0$
- $x^{2} - 16 x + 64 \geq 0$
- $x^{2} - 5 x + 6 \leq 0$
- $x^{2} - 18 x + 80 > 0$
- $x^{2} - 4 x + 3 < 0$$
