Introduction to Inequalities
Inequality tells us about the relative size of two
values.
Mathematics is not always about
"equals"! Sometimes we only know that something is bigger or smaller.
Greater or Less Than
The two most common inequalities
are:
|
Symbol
|
Words
|
Example Use
|
|
>
|
greater than
|
5 > 2
|
|
<
|
less than
|
7 < 9
|
They are easy to remember: the
"small" end always points to the smaller number, like this:
Greater Than Symbol: BIG
> small
Example: Alex and Chris have a race, and Chris wins!
What do we know?
- We
don’t know how fast they ran, but we do know that Chris was faster than
Alex:
Chris
was faster than Alex
We can write that down like this:
c > a
Where “c” means how fast Chris was, “>” means “greater than”, and “a”
means how fast Alex was.
In this example we called it “Inequalities” because they are not equal.
...Or Equal To!
We can also have inequalities that include
"equals", like:
|
Symbol
|
Words
|
Example Use
|
|
≥
|
greater than or
equal to
|
x ≥ 1
|
|
≤
|
less than or
equal to
|
y ≤ 3
|
Example: You must be 13 or older to watch a movie.
The “Inequality” is between your age
and the age of 13.
Your age must be “greater than or
equal to 13”, which is written:
Age ≥ 13
Solving
Inequalities
Sometimes we need to solve Inequalities like
these:
|
Symbol
|
Words
|
Example
|
|
>
|
greater than
|
x + 3 > 2
|
|
<
|
less than
|
7x < 28
|
|
≥
|
greater than or equal to
|
5 ≥ x
- 1
|
|
≤
|
less than or equal to
|
2y + 1 ≤ 7
|
Solving
Our aim is to have x (or whatever the
variable is) on its own on the left of the inequality sign:
|
Something like:
|
|
x < 5
|
|
or:
|
|
y ≥ 11
|
We call that “Solved”.
How to Solve?
Solving inequalities is very like solving
equations, we do most of the same things but we must also pay attention to the
direction of the inequality.
Direction:
Which way the arrow "points"
Some things we do will change the
direction!
< would become >
> would become <
≤ would become ≥
≥ would become ≤
Safe
Things to Do!
These
are things we can do without affecting the direction of the
inequality:
- Add
(or subtract) a number from both sides
- Multiply
(or divide) both sides by a positive number
- Simplify
a side
Example: 3x < 7+3
We can simplify 7+3
without
affecting the inequality:
3x
< 10
But these things will change the
direction of the inequality ("<" becomes ">"
for example):
- Multiply
(or divide) both sides by a negative number
- Swapping
left and right hand sides
Example: 2y+7
< 12
When we swap the left and right hand
sides, we must also change the direction of the inequality:
12 > 2y+7
Here
are the details:
Adding
or Subtracting a Value
We
can often solve inequalities by adding (or subtracting) a number from both
sides (just as in Introduction to Algebra), like this:
Solve: x
+ 3 < 7
If
we subtract 3 from both sides, we get:
x + 3 - 3 < 7 - 3
x < 4
And
that is our solution: x < 4
In
other words, x can be any value less than 4.
What
did we do?
|
We
went from this:
To
this:
|
|
 |
|
x+3
< 7
x
< 4
|
And
that works well for adding and subtracting,
because if we add (or subtract) the same amount from both sides, it does not
affect the inequality.
Example: Alex has more coins than Billy. If both Alex
and Billy get three more coins each, Alex will still have more coins than
Billy.
What
If I Solve It, But "x" Is On The Right?
No
matter, just swap sides, but reverse the sign so it still
"points at" the correct value!
Example: 12 < x + 5
If
we subtract 5 from both sides, we get:
12 - 5 < x + 5 - 5
7 < x
That
is a solution!
But
it is normal to put "x" on the left hand side so let us flip sides
(and the inequality sign!):
x
> 7
Do
you see how the inequality sign still "points at" the smaller value
(7)?
And
that is our solution: x > 7
Note:
"x" can be on the right, but people usually like to
see it on the left hand side.
Multiplying
or Dividing by a Value
Another
thing we do is multiply or divide both sides by a value (just as in Algebra -
Multiplying).
But
we need to be a bit more careful (as you will see).
Positive
Values
Everything
is fine if we want to multiply or divide by a positive number:
Solve: 3y
< 15
If
we divide both sides by 3 we get:
3y/3 < 15/3
y < 5
And
that is our solution: y < 5
Negative
Values
When we multiply or divide by a negative
number we must reverse the inequality.
Why?
Well,
just look at the number line!
For example, from 3 to 7 is an increase, but
from -3 to -7 is a decrease.
|
-7
< -3
|
7
> 3
|
See how the inequality sign reverses (from <
to >)?
Let us try an example:
Solve: -2y
< -8
Let us divide both sides by -2 ... and reverse
the inequality!
-2y < -8
-2y/-2 > -8/-2
y > 4
And that is the correct solution: y
> 4
(Note that I reversed the inequality on
the same line I divided by the negative number.)
So, just remember:
When multiplying or dividing by a negative
number, reverse the inequality.
Multiplying or Dividing by Variables
Here is another (tricky!) example:
Solve: bx
< 3b
It seems easy just to divide both sides
by b, which would give us:
x < 3
But wait, if b is negative we
need to reverse the inequality like this:
x
> 3
But we don't know if b is positive or negative,
so we can't answer this one!
To help you understand, imagine replacing b with 1 or -1 in
that example:
- if b
is 1, then the answer is simply x < 3
- but
if b is -1, then we would be solving -x < -3,
and the answer would be x > 3
So:
Do not try dividing by a variable to
solve an inequality (unless you know the variable is always positive, or always
negative).
A Bigger Example
Solve: (x-3)/2
< -5
First, let us clear out the "/2" by
multiplying both sides by 2.
Because we are multiplying by a positive
number, the inequalities will not change.
(x-3)/2 ×2 <
-5 ×2
(x-3) < -10
Now add 3 to both sides:
x-3 + 3 < -10 + 3
x < -7
And that is our solution: x
< -7
Two Inequalities At Once!
How do we solve something with two inequalities
at once?
Solve:
-2 < (6-2x)/3 < 4
First, let us clear out the "/3" by
multiplying each part by 3:
Because we are multiplying by a positive
number, the inequalities will not change.
-6 < 6-2x < 12
-12 < -2x < 6
Now multiply each part by -(1/2).
Because we are multiplying by a negative number,
the inequalities change direction.
6 > x > -3
And that is the solution!
But to be neat it is better to have the smaller
number on the left, larger on the right. So let us swap them over (and make
sure the inequalities point correctly):
-3 < x < 6
Good, it is clear and easy to understand :)
ReplyDeleteThank you very much. Really appreciate that.
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