Test: Linear Inequalities One Variable

Let x be the smaller of the two consecutive even positive integers, then the other even integer is x + 2. 
 Given x < 10  and x + (x + 2) > 11.  
⇒ x < 10, and 2x + 2 > 11.  
⇒ x < 10, 2x > 9 
⇒ x < 10, x > 9/2 
⇒ 10 < x > 9/2   
∴ the required parity even integers is (6, 8)

Quadratic inequalities can be of the following forms:
ax² + bx + c > 0
ax² + bx + c ≥ 0
ax² + bx + c < 0
ax² + bx + c ≤ 0

Let marks required be x 
= (45 + 62 + x)/3 = 60
45+62+ x = 60*3
107 + x = 180 
x = 180 - 107
x = 73

when x=3 
the 12(3) = 36 which is greater than 30

The given inequality is 5x– 3 < 7
=> 5x – 3 + 3 < 7 + 3                             [3 is added both sides]
=> 5x < 10
=> x < 10/5
=> x < 2
When x is a real number, the solutions of the given inequality are given by x < 2, i.e., all real numbers x which are less than 2.

-5x + 2 < 7x - 4
2 + 4 < 7x + 5x
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}

3x - 7> x + 3
2 x > 10
x > 5
so x- coordinate should be > 5

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

As we move to right side on the number line the value increases. i.e. To the right of the point (-3,0)

4x + 3 < 5x + 7
subtract 4 both sides,
4x + 3 - 3 < 5x + 7 - 3
⇒ 4x < 5x + 4
subtract ' 5x ' both sides ,
[ equal number may be subtracted from both sides of an inequality without affecting the sign of inequality]
4x - 5x < 5x + 4 - 5
-x < 4
now, multiple with (-1) then, sign of inequality change .
-x.(-1) > 4(-1)
x > -4
hence, x€ ( -4 , ∞)

Two less than 5 times a number is greater than the third multiple of the number.
5x - 2 >3x
5x - 3x > 2
2x > 2
x > 1