Topic 2: Inequalities & Exercise

INEQUALITIES

Definition of inequalities

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.
   
Inequalities tells us about the relative size of two values. Mathematics is not always about "equal" but sometimes we only know that somethings is "bigger or smaller".

 -.

Rule of Inequalities

1) a < b means a is a smaller number than b.

2) a ≤ b means a is a smaller number than b or they are equal.

3) a > b means a is a larger number than b.

4) a ≥ b means a is a larger number than b or they are equal.

Important thing!!! 👇

Example 1: 

3x < 7+3

We can simplify 7 + 3 without affecting the inequality:

3x < 10

But these things 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 2:

2y + 7 < 12
When we swap the left and right hand sides, we must also change the direction of the inequality:
12 > 2y + 7


Example 3:

Solve: 2x + 3 ≤ 15

Solution: 2x ≤ 15 - 3

why its (-3) because in front the number 3 is (+) so if the sign bring to backwards it's become (-) sign. 

2x ≤ 12
x ≤ 12/2
x ≤ 6

Adding or Subtraction a value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in introduction on 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.

 

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 1: 3y < 15

If we divide both sides by 3 we get:

3y/3 < 15/3

y < 5

And that is our solution: y < 5

Example 1:

Solve 2: -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 1: bx < 3b

It seems easy just to divide both sides by b, which gives 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 the example of bx < 3b:

• if b is 1, then the answer is x < 3
• but if b is -1, then we are solving -x < -3, and the answer is x > 3

The answer could be x < 3 or x > 3 and we can't choose because we don't know b.
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

Now subtract 6 from each part:

-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

Below is the video to show you how to solve the inequalities

Exercise

1) solve the inequalities: 2 + 6x < 4

2) 8 - 2x > 2

3) Ali has $10, while Azman has only $3. how do you do the sign?