Hi Bharti,
solving inequalities is about one thing: sign changes. Find all the points at
which there are sign changes - we call these points critical values. Then determine
which, if any, of the intervals bounded by these critical values result in a solution. The
solution to the inequality will consist of the set of all points contained by the solution
intervals.
Which of the following inequalities have a finite range of values of "x" satisfying them?
A. x2 + 5x + 6 > 0
B. |x + 2| > 4
C. 9x - 7 < 3x + 14
D. x2 - 4x + 3 < 0
E. (B) and (D)
A. x2 + 5x + 6 > 0 =>(x+2)(x+3)>0
Here the critical points are when the expression (x+2)(x+3) = 0, i.e. at x = -2 and -3.
Let us check if the intervals bounded by these critical values result in a solution.
If x<-3
Expression (x+2)*(x+3)>0 (if x =-4, (-4+2)*(-4+3) >0) - you can stop solving the problem here because x<-3 is the solution and consists of an infinite solutions.
If -3
<x
<-2,
Expression (x+2)*(x+3)
<0 (if x =-2.5, (-2.5+2)*(-2.5+3)<0)
If x<-2
Expression (x+2)*(x+3)>0 (if x =0, (0+2)*(0+3) >0).
Solution for the inequality is x<-3(-Infinity to -3 in school terms) or x>-2(2 to infinity in school terms), which is NOT a finite set of real numbers.
B
|x + 2| > 4.
If|Y|>a then the solution is Y > a or Y <-a
So,
x + 2 > 4 or x + 2 < -4
x > 2 (2 to Infinity in school terms)or x < -6 (-Infinity to -6 in school terms)is the solution, which is NOT a finite set of real numbers.
C
9x - 7 < 3x + 14
6x < 21
x < 7/2 (-Infinity to 7/2 as learnt at school)is the solution, which is NOT a finite set of real numbers.
D
You need not check for this option because this is the only one left but let me check for your benefit.
x2 - 4x + 3 < 0 , (x-1)(x-3)<0
Here the critical points are when the expression (x-1)(x-3) = 0, i.e. at x = 1 and 3.
Let us check if the intervals bounded by these critical values result in a solution.
If x
<1
Expression (x-1)(x-3)
>0 (if x=0, (0-1)*(0-3) >0)
If 1<x<3,
Expression (x-1)(x-3)<0 (if x=2, (2-1)*(2-3)<0)
If x
>3
Expression (x-1)(x-3)
>0 (if x=4, (4-1)*(4-3) >0)
1<x<3 is the solution, which is a finite set.
Option
D, is the answer!
