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LSC2: INEQUALITIES AND REGIONS

This unit is about inequalities and regions

INEQUALITIES AND REGIONS

Introduction to Inequalities

Inequality tells us about the relative size of values.

Mathematics is not always about “equals”, sometimes we only know that something is greater or less than.

Example: Alex and Billy have a race, and Billy wins!

What do we know?

We don’t know how fast they ran, but we do know that Billy was faster than Alex:

Billy was faster than Alex

We can write that down like this:

b > a

(Where “b” means how fast Billy was, “>” means “greater than”, and “a” means how fast Alex was)

We call things like that inequalities (because they are not “equal”)

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 sign

Greater Than Symbol: BIG > small

 

soccer ball

Example: Alex plays in the under 15s soccer. How old is Alex?

We don’t know exactly how old Alex is, because it doesn’t say “equals”

But we do know “less than 15”, so we can write:

Age < 15

The small end points to “Age” because the age is smaller than 15.

… 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”.

Example: x + 2 > 12

Subtract 2 from both sides:

x + 2 − 2 > 12 − 2

Simplify:

x > 10

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.

greater than sign
Direction: Which way the arrow “points”

Some things can change the direction!

< becomes >

> becomes <

≤ becomes ≥

≥ becomes ≤

Safe Things To Do

These things do not affect 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 do 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:

Example: 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:

 number line inequality x+3 < 7

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:

Example: 3y < 15

If we divide both sides by 3 we get:

3y/3 < 15/3

y < 5

And that is our solution: y < 5

What are inequalities on a number line?

 

Inequalities on a number line allow us to visualise the values that are represented by an inequality.

 

To represent inequalities on a number line we show the range of numbers by drawing a straight line and indicating the end points with either an open circle or a closed circle.

 

An open circle shows it does not include the value.

A closed circle shows it does include the value.

 

E.g.

 

Inequalities on a number line image 1 1

 

The solution set of these numbers are all the real numbers between 1 and 5.

As 1 has an open circle, it does not include ‘1’ but does include anything higher, up to and including 5 as this end point is indicated with a closed circle.

 

We can represent this using the inequality

 

We can also state the integer values (whole numbers) represented by an inequality.
In this example, the integers 2, 3, 4 and 5 are all greater than 1 but less than or equal to 5.

 

The solution set can represent all the real numbers shown within the range and these values can also be negative numbers.

How to represent inequalities on a number line

 

In order to represent inequalities on a number line:

  1. Identify the value(s) that needs to be indicated on the number line.
  2. Decide if it needs an open circle or a closed circle;
    < or > would need an open circle
    \leq or \geq would need a closed circle.
  3. Indicate the solution set with a straight line to the left hand side or right hand side of the number or with a straight line between the circles.

 

E.g.

 

Represent x < 3 on a number line

An open circle needs to be indicated at ‘3’ on the number line.

 

Inequalities on a number line image 2 1

 

As x < 3 is ‘x is less than 3’, the values to the left hand side of the circle need to be indicated with a line.

 

Inequalities on a number line image 3 1

 

E.g.

 

Represent 2<{x}\leq{6} on a number line.

 

An open circle needs to be indicated above ‘2’ and a closed circle needs to be indicated above ‘6’.

 

Inequalities on a number line image 4 1

 

Then draw a line between the circles to indicate any value between these circles.

 

Inequalities on a number line image 5 1

Representing inequalities on a cartesian plane

 

 

 

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