Notes on inequalities

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Additional Mathematics
Algebra: Techniques
Section 1: Linear and quadratic inequalities
Notes and Examples
These notes contain subsections on
Linear inequalities
Quadratic inequalities
Linear inequalities
You already know that y = 2x + 1 is a straight line but what happens if we need
to look at y 2x + 1 or y 2x + 1. If you draw the line and then move a ruler
upwards parallel to it, the values of x + 1 will not change but the corresponding
y values will be bigger. Similarly if you move the ruler down from the line, x + 1
will stay the same but the y values will be smaller; i.e. points above the line
will satisfy y 2x + 1 and those below the line will satisfy y 2x + 1. To solve
inequalities we follow the same procedure as for solving equations except
when we multiply or divide by a negative number. In this case the direction of
the inequality reverses. E.g. 4 2 but if we multiply by ­ 1, 4 is not greater
than 2, but less than 2.
It is best to try and avoid dividing by a negative as in the following example.
Add x to both sides
7 15 ­ x
7 + x 15 Take 7 away from both sides
Doing the inequality in this way meant that we weren't left with: ­ 8 x and
the need to multiply or divide by ­ 1.
Example 1
(i) 2x ­ 3 9
(ii) 6y + 1 4y + 9
(iii) 11 3x + 5 20
(i) 2x ­ 3 9
2x 12
x6 Collect like terms i.e. take
4y away from both sides
(ii) 6y + 1 4y + 9
2y + 1 9
Take 1 from both sides
© MEI, 24/07/09 1/3

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AM Algebra Techniques 1 Notes and Examples
(iii) 11 3x + 5 20
11 3x + 5 and 3x + 5 20
Treat as two halves
6 3x and 3x 15
2 x and x 5
Therefore, 2x
You can see further examples using the Flash resource Linear inequalities.
You can test yourself using the interactive questions Solving linear
Quadratic inequalities
With quadratic inequalities we either need to draw a graph of the function or
we could analyse the situation.…read more

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AM Algebra Techniques 1 Notes and Examples
Example 3
(i) a2 + 6 a + 5 0
(ii) p2 3p 10
Solutions Factorise Two quantities multiply to give
a negative; this implies that
(i) a2 + 6 a + 5 0 they are of different signs.
(a + 5)(a + 1) 0
Either a + 5 0 and at the same time a + 1 0 i.e.…read more




A good set of worked examples and explanation for solving linear and quadratic inequalities.  A written mistake in the initial paragraph which should say that the value of 2x +1 remains the same (rather than just x+1),  y is greater above the line or y is less below the line...  This may confuse if not corrected.

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