# Inequalities

Notes on inequalities

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Algebra: Techniques
Section 1: Linear and quadratic inequalities
Notes and Examples
These notes contain subsections on
Linear 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.
7 15 ­ x
7 + x 15 Take 7 away from both sides
x8
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
Solve:
(i) 2x ­ 3 9
(ii) 6y + 1 4y + 9
(iii) 11 3x + 5 20
Solutions
(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

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AM Algebra Techniques 1 Notes and Examples
y8
(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
inequalities.
With quadratic inequalities we either need to draw a graph of the function or
we could analyse the situation.…read more

### Page 3

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AM Algebra Techniques 1 Notes and Examples
Example 3
Solve:
(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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