Transcript Revision Linear Inequations

Revision
Linear Inequations
Algebraic and Graphical Solutions.
By I Porter
Introduction
An inequation is formed when two mathematical statements have an unequality
sign between them.Comon inequality signs:
> is greater than
≥ is greater than or equal to
< is less than
≤ is less than or equal to
Inequations can have an infinite number of solutions.
Solving inequations makes use of the following axioms of inequality for real
numbers a, b and c.
If a > b , then
5. ac < bc if c < 0
a b

6.
if c < 0
c c
1. a + c > b + c
2. a - c > b - c
3. ac > bc if c > 0
a b

4.
if c > 0
c c

Similar axioms also apply for a < b.

Solving Inequalities
Inequations may be simplified by:
1. adding the same number to both sides.
i.e. 10 > 3, then 10 + 2 > 3 + 2
2. subtracting the same number to both sides.
i.e. 10 > 3, then 10 - 2 > 3 - 2
i.e. 10 > 3, then 10 x 2 > 3 x 2
10 3

i.e. 10 > 3, then
4. dividing both sides by the same positive number.
2 2
In all cases above, the direction of the inequality remains the same.
Also, the above statements apply when ‘>’ is replaced with ‘<‘.

Special Cases
3. multiplying both sides by the same positive number.
The inequality sign must be reversed when:
1. multiplying both sides by the same negative number.
2. dividing both sides by the same negative number.
i.e. 10 > 3, but 10 x -2 < 3 x -2
10
3
i.e. 10 > 3, then
<
2 2


Graphical Solutions
Algebra
Graphical Number Line Solution
x>4
2
3
4
5
6
-7
-6
-5
-4
-3
-12
-11
-10
-9
-8
16
17
18
19
20
x < -5
x ≥ -10
x ≤ 18
x
x
x
x
Note: ‘>’ and ‘< ‘ use and open circle.
Note: ‘≥’ and ‘≤‘ use and closed (dot) circle.
You also only need to write in three (3) numbers to indicate location and order.
Examples
Inequations are solve exactly the same way as equations, with two exception as stated by
the axioms (5) and (6). [ reverse the inequality when x or  by a negative number ]
Solve and graph a solution for the following:
Add 5 to both sides.
a) 4x - 5 < 23
6
b) 4(2 - x) ≥ 3x + 14
Expand Brackets.
4x < 28
Divide both sides by 4.
8 - 4x ≥ 3x + 14
Subtract 3x from both sides.
x<7
Open circle, arrow left.
8 - 7x ≥ 14
Subtract 8 from both sides.
7
8
Divide both sides by -7.
- 7x ≥ 6
6
x
7
x

1
6
7
Reverse inequality sign.
Closed circle, arrow left.
5
7
x



Always move ALGEBRA to the LEFT SIDE
Examples: Solve and graph a solution for the following:
x 1 x 1

5
3
a)
3(x 1)  5(x 1)
Subtract 5x from both sides.
2x  3  5

Add 3 to both sides.
2x  8
x  4




-5
-4
Divide both sides by -2.
Reverse inequality sign.
Closed circle, arrow right.
-3
x
Add 3 to both sides.
2  2x 10
Expand brackets.
3x  3  5x  5

5  2x  3  7
b)
Cross multiply denominators.
Divide both sides by 2.
1 x  5



-2
-1
0
1
Always move ALGEBRA to the LEFT SIDE
2
3
4
5
6
7
Exercise: Solve and graph a solution for each the following:
2) 7x  3(2x 1)
1) 2x  5  5
x3
x  5
-6
-5
-4
 x

2
x  3
3
4
 x
-4

x  5 5x  3

2
6
4)
x  6
-7

3) 2x  7  3x 10
-6
6)  3 
5 x7

-5

x
5
6
7
2x 1
3
3
4  x  5

4
-2

5) 22  5x  3  32
x
-3
8
x
-4

0
5
x