Numberphile: 07/07/16

INEQUALITIES

Definition

An inequality says that two values are not equal. a ≠ b says that a is not equal to b. There are other special symbols that show in what way things are not equal

a < b says that a is less than b
a > b says that a is greater than b
(those two are known as strict inequality)

a ≤ b means that a is less than or equal to b
a ≥ b means that a is greater than or equal to b

=	When two values are equal we use the "equals" sign	Example: 2+2 = 4
≠	When two values are definitely not equal we use the "not equal to" sign	Example: 2+2 ≠ 9
<	When one value is smaller than another we use a "less than" sign	Example: 3 < 5
>	When one value is bigger than another we use a "greater than" sign	Example: 9 > 6

Another Example

10 > 5

"10 is greater than 5"

Or the other way around:

5 < 10

"5 is less than 10"

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

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

2y+7 < 12

When we swap the left and right hand sides, we must also change the direction of the inequality:

12 > 2y+7

Multiplying or Dividing by a Value

Solve

3y < 15

If we divide both sides by 3 we get:

3y/3 < 15/

y < 5

And that is our solution: y <

Negative Values

When we multiply or divide by a negative number we must reverse the inequality

For Example

From 3 to 7 is an increase, but from -3 to -7 is a decrease

-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

When multiplying or dividing by a negative number, reverse the inequality

Multiplying or Dividing by Variables

Solve

bx < 3b
It seems easy just to divide both sides by b, which would give us:

x < 3

if b is negative we need to reverse the inequality like this:

x > 3

(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