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Some inequalities contain more information and need further development.


Solve the inequality

x - 10 < 2x - 2 < x


The method is the same as before but now there are two inequalities to solve, i.e. x - 10 < 2x - 2 and 2x - 2 < x. The first of these is solved in the left-hand column, the second in the right-hand column.

Here are some examples for you to practise on.


Find the solution to each of the following inequalities.


(a) Here there are two inequalities to be solved, - 3 3 x and 3 x 18. The first of these is

- 3 3x , divide both sides by 3

- 1 x.

The second is

3x 18 , divide both sides by 3

x 6 . In both of the above inequalities the divisor is 3 , which is positive, so the division does not reverse the inequalities.

The solution to the inequality is thus - 1 x 3.

(b) There are two inequalities here, 10 2 x and 2 x x + 9 .

The first is

10 2 x , divide both sides by 2

5 x .

The second is

2 x x + 9 , subtract x from both sides

x 9 .

The solution to the inequality is 5 x 9 .

(c) Here there are two inequalities, x < 3x - 1 and 3x - 1 < 2x + 7. They are solved as follows.

x < 3x - 1, adding 1

x + 1< 3x, substracting x

1 < 2x, dividing by 2

1/2 < x.

3x - 1 < 2x + 7, adding 1

3x < 2x + 8, substracting 2x

x < 8.

The solution to the inequality is 1/2 < x < 8.

(d) The two inequalities in this case are 2 x - 7 < 8 and 8 < 3x - 11. The solution to each is

2 x - 7 < 8, add 7

2x < 15, divide by 2

x < 15/2.

8 < 3x - 11, add 11

19 < 3x, divide by 3

19/3 < x.

The solution to this is 19 /3 < x < 15/2.

To end this section try the short quiz below.


Which prime numbers satisfy the inequality.

0 2w - 3 w + 8 ?

(a) 5,7,11,13 (b) 2,5,11,17 (c) 2,3,5,7 (d) 3,7,11,13


As in previous cases there are two inequalities to be solved, 0 2 w - 3 and 2 w - 3 w + 8.

The solution to each of these is

0 2 w - 3 , add 3

3 2 w , divide by 2

3 /2 w .

2 w - 3 w + 8 , add 3

2 w w + 11 , subtract w

w 11 .

so 3 / 2 w 11. The prime numbers in this range are 2,3,5,7,11 which includes ALL those of part (c) but not all of the other choices on offer.

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