SOLVING INEQUALITIES - ANSWERS

1.  x + 7 < 10

Subtract 7 from both sides
x < 10 - 7
i.e. x < 3

2.  6 - 3x > 12

Add 3x to both sides
6 > 12 + 3x

Subtract 12 from both sides
-6 > 3x

Divide both sides by 3
-2 > x

3.  3x + 7 > 2(x + 3)

Multiply out the brackets on the right
3x + 7 > 2x + 6

Subtract 2x from both sides
x + 7 > 6

Subtract 7 from both sides
x > -1

4.  x2 - 9 < 0

This is an example of a quadratic inequality. If you factorise the left (notice that it is the difference of 2 squares) you get:
(x + 3) (x - 3) < 0

Now, change the < to = and solve the quadratic
(x + 3) (x - 3) = 0

Gives x = -3 or x = 3 (these are called the critical values)

Now as the inequality is < we can say:
-3 < x < 3

(note: if the inequality was £, the answer would be -3 £  x  £ 3)

5.  For 3n + 2 < 10, find the greatest      integer value of n
6. Write down an inequality satisfied 
     by  the integers: -2, -1, 0, 1, 2		 -2  £  x  £  2 
7.  1 £ 3n + 4 £ 10

When there is more than one equality, split them up and then join them in the answer
1 £ 3n + 4              3n + 4 £ 10
-3 £ 3n                          3n £ 6
-1 £ n                              n £ 2

Hence -1 £ n £ 2

8. Find all possible integer values of x
    for the inequality:

    4 < 5x - 1 £ 14

4 < 5x - 1              5x - 1 £ 14
5 < 5x                         5x £ 15
1 < x                             x £ 3

1 < x £ 3

x is greater than 1 so we can't include this, and x is less than or equal to 3, so the integer solutions are:

x = 2, 3