Test: Graphical Solutions - JEE MCQ

Let x be the smaller of the two consecutive even positive integers .
Then the other integer is x+2.
Since both the integers are smaller than 10,x<10 ....(1)
Also the sum of the two integers is more than 11.
x+(x+2)>11
⇒ 2x+2>11
⇒ 2x>11−2
⇒ 2x>9
⇒ x>9/2
⇒ x>4.5....(2)
From (1) and (2) we obtain 4.5>x>11
Since x is an even number, x can take the values 6,8 and 10.
Thus the required possible pairs are (6,8).

|3x - 1| + 1 < 3
|3x -1| < 2
Opening mod, we get
3x - 1 < 2,  -3x + 1 > 2
3x < 3,   -3x > 1
x < 1,   x > -1/3
-1/3 < x < 1

No of attempts = 3
Average =  (45+62+x)/3=60
x=73

12x > 30
x > 30/12
x > 2.5
x is an integer. So, minimum value of x is 3.

 -5x + 2 < 7x - 4
6 < 12x
x > 1/2

We have 5x−3<3x+1
⇒5x−3+3<3x+1+3
⇒5x<3x+4
⇒5x−3×<3x+4−3x
⇒2x<4⇒x<2
When x is an integer the solutions of the given inequality are {.............,−4,−3,−2,−1,0,1}
Hence {x / xεZ, x<2}

-4x+1≥0 and 3-4x<0

 1 ≥ 4x and 3 < 4x

1/4 ≥ x  and 3/4 < x

so x ∈ (-∞,1/4] U (3/4 , ∞ )

(2-3x)/5 ≤ (-x-6)/2
By cross multiply we get
4-6x ≤ -5x-30
-x ≤ -34
x ≥ 34

By euler’s formula the relation between vertices(n), edges(q) and regions(r) is given by n-q+r=2.

5x+6<2x-3
5x-2x < -3-6
3x < -9
x<-3

4x+3<5x+7
4x+3−7<5x+7−7
4x−4<5x
4x−4−4x<5x−4x
−4<x
Thus, the solution set of the given inequality is (−4,∞).

- 6x < 24
=> 6x < -24
x < - 4

X/3-x/2 > 1
-x/6 > 1
x/6 
x < -6
therefore x belongs to the range (-∞,-6)

24x < 100
⇒ x < 100/24
⇒ x < 25/6
 It is evident that 1,2,3 and 4 are the only natural numbers less than 25/6,
Thus  when x is a natural number ,the solutions of the given inequality are 1,2,3 and 4.
Hence, in this case, the solution set is {1,2,3,4}.