**Solving Linear Inequalities**

Example 1

Example 2

Example 3

Solving double inequalities.

Solve the inequality:

5 < 3x - 1 < 14

1) Get rid of -1 in the middle by +ing +1 to both side of the inequality

2) Simplify the inequality

3) To get x on it's own, divide both sides

by 3.

Example 4

Find the set of values of x for which:

3(x-5) > 5-2(x-8)

5 < 3x - 1 < 14

+1

: 5 + 1 < 3x - 1 + 1 < 14 + 1

simplify

: 6 < 3x < 15

÷ by 3

: 2 < x < 5

Solve the inequality:

2x + 4 > 8x - 2

1) Get x terms on one side and numbers on other

2) Simplify the inequality

3) Divide both sides by the number in front

of x.

Note: when you multiply or divide an inequality by a negative number, the inequality sign needs to be changed to it's opposite. Example:

2x + 4 > 8x - 2

collect like terms

: 4 + 2 > 8x - 2x

Simplify

: 6 > 6x

÷ by 6:

1 > x

12 - 3x < 27

- 12

: -3x < 15

÷ by -3

: x > -5

Find the set of values of x for which: 4x+7>3 & 17<11+2x by illustrating it on a number line.

4x + 7 > 3

-7:

4x > -4

÷ by 4:

x > -1

17 < 11 + 2x

-11

: 6 < 2x

÷ by 2

:

3 < x

x > 3

Using a number line, it is easy to see that the two sets of value overlap where x>3

3(x-5) > 5-2(x-8)

Multiply out

: 3x -15 > 5 -2x +16

+15

: 5x > 5 + 16 + 15

Simplify

: 5x > 36

÷ by 5

: x > 7.2

**Solving Quadratic Inequalities**

Example 1

Find the set of values of x for which i)

x -4x-5<0

, ii)

x -4x-5>0

and draw sketches to show

these.

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i) x -4x-5 < 0

x -4x-5 = 0

factorise: (x+1)(x-5) = 0

x=-1 x=5

so: -1 < x < 5.

x -4x-5 < 0

x -4x-5 = 0

factorise: (x+1)(x-5) = 0

x=-1 x=5

so: x < -1

x > 5

ii)

note: in both these cases the x is positive, so the graphs would be a +ve parabola.

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Example 2

Find the set of values of x for which

3 - 5x - 2x < 0

and sketch a graph to show this.

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critical values

3 - 5x - 2x < 0

change into equation

: 3 - 5x - 2x = 0

take everything to the other side

: 2x +5x - 3 = 0

factorise

: (2x-1) (x+3)

critical values

: x=1/2 or x=-3

so

: x < -3

x > 1/2

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The 4 inequalities

In an equation the '=' sign means that the two sides are identical. But what happens when the two sides are not identical?

If this is the case, inequalities are used to show the relationship between the two sides.

THE END

Note: since the coefficient of x is negative, the graph is a -ve parabola ('up-side down u shape')

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