Download How to solve inequalities.

How to solve inequalities.
There are a number of different types of inequalities that we can solve the following notes show the
method used to solve each type of inequality problem.
Type 1: “Linear inequalities” The method we use to solve “linear inequalities” is the same method
that is used to solve any linear equation. We will therefore add, subtract, multiply or divide both
sides until you get a single variable x on its own. The main differences in solving an inequality is
that if you multiply or divide both sides by a negative number then the inequality symbol will
reverse direction and the second difference is that your solution will not be a single value of x but
an interval of values. For example, when you solve an equation you typically get x = 2 but when
you solve an inequality the solution will be an interval of values such as x > 2 or x 2.
Also notice that when we give a solution we do not just stop and just say that the solution is x 4
but we also give the solution in interval notation. You must express all solutions to inequalities in
interval notation unless you are told otherwise.
Example 1:
2x + 3 <
2x
<
x
<
21
18
9
Example 2:
4x – 3
4x
x
17
20
5
Example 3:
5 – 2x
– 2x
x
13
8
–4
Example 4:
4 – 3x
– 3x
2x
x
12 – 5x
8 – 5x
8
4
Example 5:
4 – 3x >
– 3x >
2x
>
x
– 12 – 5x
– 16 – 5x
– 16
–8
Example 6:
10 – 3(x – 6)
10 – 3x + 18
28 – 3x
– 3x
– 4x
x
>
>
>
>
<
(
x–2
x–2
x – 30
– 30
7.5
Notice the reversing of the symbol
into
Notice the reversing of the symbol
into
)
Type 2: “Simple Quadratic inequalities” The method we use to solve “Simple linear inequalities”
is to Foil out the expressions and then to remove the x2 terms when this is done we will be left with
a straight forward linear inequality and we will solve it in the usual way.
To work with Simple Quadratic inequalities we need to remember the technique called F.O.I.L
First
Outer Inner Last
Outer
First
For example
(2x + 11)(x – 3) =
Inner
2x2 – 6x + 11x
F
O
I
– 33
L
Last
Example 1:
Example 2:
Example 3:
(x +3)(x – 6)
x2 – 6x + 3x – 18
x2 – 3x – 18
– 3x – 18
– 3x
–x
x
<
<
<
<
<
<
>
x(x – 15 )
x2 – 15x
x2 – 15x
x2 – 15x
– 15x
– 9x
x
(2x +1)(3x – 1)
6x2 – 2x + 3x – 1
6x2 + x – 1
x –1
x
12x
x
<
<
<
<
<
<
<
(x +1)(x – 3)
x2 – 3x + x – 3
x2 – 2x – 3
– 2x – 3
– 2x + 15
15
– 15
Using F.O.I.L
Collect like terms
Subtract x2 from both sides
Add 18 to both sides
Add 2x to both sides
Divide both sides by – 1
(x – 3)2
(x – 3)(x – 3)
x2 – 3x – 3x + 9
x2 – 6x + 9
– 6x + 9
9
–1
Use the Distributive law
Using F.O.I.L
Collect like terms
Subtract x2 from both sides
Add 6x to both sides
Divide both sides by – 9
(6x +1)(x – 2)
6x2 – 12x + x – 2
6x2 – 11x – 2
– 11x – 2
– 11x – 1
–1
Using F.O.I.L
Collect like terms
Subtract 6x2 from both sides
Add 1 to both sides
Add 11x to both sides
Divide both sides by 2