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 inIntroduction 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

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

Summary

• Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
• But these things will change direction of the inequality:
• Multiplying or dividing both sides by a negative number
• Swapping left and right hand sides
• Don't multiply or divide by a variable (unless you know it is always positive or always negative)
• 
  

Equal, Greater or Less Than

As well as the familiar equals sign (=) it is also very useful to show if something is not equal to (≠) greater than (>) or less than (<)

These are the important signs to know:

=	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

Less Than and Greater Than

The "less than" sign and the "greater than" sign look like a "V" on its side, don't they?

To remember which way around the "<" and ">" signs go, just remember:

• BIG > small
• small < BIG

The "small" end always points to the smaller number, like this:

Greater Than Symbol: BIG > small

Example:

10 > 5

"10 is greater than 5"

Or the other way around:

5 < 10

"5 is less than 10"

Do you see how the symbol "points at" the smaller value?

... Or Equal To ...

Sometimes we know a value is smaller, but may also be equal to!

Example, a jug can hold up to 4 cups of water.

So how much water is in it?

It could be 4 cups or it could be less than 4 cups: So until we measure it, all we can say is "less than or equal to" 4 cups.

To show this, we add an extra line at the bottom of the "less than" or "greater than" symbol like this:

The "less than or equal to" sign:		≤
The "greater than or equal to" sign:		≥

All The Symbols

Here is a summary of all the symbols:

Symbol	Words	Example Use
=	equals	1 + 1 = 2
≠	not equal to	1 + 1 ≠ 1
>	greater than	5 > 2
<	less than	7 < 9
≥	greater than or equal to	marbles ≥ 1
≤	less than or equal to	dogs ≤ 3

Why Use Them?

Because there are things we do not know exactly ...

... but can still say something about.

So we have ways of saying what we do know (which may be useful!)

Example: John had 10 marbles, but lost some. How many has he now?

Answer: He must have less than 10:

Marbles < 10

If John still has some marbles we can also say he has greater than zero marbles:

Marbles > 0

But if we thought John could have lost all his marbles we would say

Marbles ≥ 0

In other words, the number of marbles is greater than or equal to zero.

Combining

We can sometimes say two (or more) things on the one line:

Example: Becky starts with $10, buys something and says "I got change, too". How much did she spend?

Answer: Something greater than $0 and less than $10 (but NOT $0 or $10):

"What Becky Spends" > $0
"What Becky Spends" < $10

This can be written down in just one line:

$0 < "What Becky Spends" < $10

That says that $0 is less than "What Becky Spends" (in other words "What Becky Spends" is greater than "$0") and what Becky Spends is also less than $10.

Notice that ">" was flipped over to "<" when we put it before what Becky spends - always make sure the small end points to the small value.

Changing Sides

We saw in that previous example that when we change sides we flipped the symbol as well.

This:		Becky Spends > $0	(Becky spends greater than $0)
is the same as this:		$0 < Becky Spends	($0 is less than what Becky spends)

Just make sure the small end points to the small value!

Here is another example using "≥" and "≤":

Example: Becky has $10 and she is going shopping. How much will shespend (without using credit)?

Answer: Something greater than, or possibly equal to, $0 and less than, or possibly equal to, $10:

Becky Spends ≥ $0
Becky Spends ≤ $10

This can be written down in just one line:

$0 ≤ Becky Spends ≤ $10

A Long Example: Cutting Rope

Here is an interesting example I thought of:

Example: Sam cuts a 10m rope into two. How long is the longer piece? How long is the shorter piece?

Answer: Let us call the longer length of rope "L", and the shorter length "S"

L must be greater than 0m (otherwise it isn't a piece of rope), and also less than 10m:

L > 0
L < 10

So:

0 < L < 10

That says that L (the Longer length of rope) is between 0 and 10 (but not 0 or 10)

The same thing can be said about the shorter length "S":

0 < S < 10

But I did say there was a "shorter" and "longer" length, so we also know:

S < L

(Do you see how neat mathematics is? Instead of saying "the shorter length is less than the longer length", we can just write "S < L")

We can combine all of that like this:

0 < S < L < 10

That says a lot:

0 is less that the short length, the short length is less than the long length, the long length is less than 10.

Reading "backwards" we can also see:

10 is greater than the long length, the long length is greater than the short length, the short length is greater than 0.

It also lets us see that "S" is less than 10 (by "jumping over" the "L"), and even that 0<10 (which we know anyway), all in one statement.

NOW, I have one more trick. If Sam tried really hard he might be able to cut the rope EXACTLY in half, so each half is 5m, but we know he didn't because we said there was a "shorter" and "longer" length, so we also know:

S<5

and

L>5

We can put that into our very neat statement here:

0 < S < 5 < L < 10

And IF we thought the two lengths MIGHT be exactly 5 we could change that to

0 < S ≤ 5 ≤ L < 10

An Example Using Algebra

OK, this example may be complicated if you don't know Algebra, but I thought you might like to see it anyway:

Example: What is x+3, when we know that x is greater than 11?

If x > 11 , then x+3 > 14

(Imagine that "x" is the number of people at your party. If there are more than 11 people at your party, and 3 more arrive, then there must be more than 14 people at your party now.)

Compare Numbers to 10

Hint: BIG > small and small < BIG

Notes

To remember which way around the "<" and ">" signs go, just remember:

• BIG > small
• small < BIG

The "small" end always points to the smaller number, like this:

Greater Than Symbol: BIG > small

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

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 Symbol: BIG > small

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

Comparing Values

Practice >, < and = with Compare Numbers to 10

Learn more about Inequalities at Less Than or Greater Than

Equations and Formulas

What is an Equation?

An equation says that two things are equal. It will have an equals sign "=" like this:

x

+

2

=

6

That equations says: what is on the left (x + 2) is equal to what is on the right (6)

So an equation is like a statement "this equals that"

What is a Formula?

A formula is a special type of equation that shows the relationship between different variables.

A variable is a symbol like x or V that stands in for a number we don't know yet.

Example: The formula for finding the volume of a box is:

V = lwh

V stands for volume, l for length, w for width, and h for height.

When l=10, w=5, and h=4, then:

V = 10 × 5 × 4 = 200

A formula will have more than one variable.

These are all equations, but only some are formulas:

x = 2y - 7	Formula (relating x and y)
a2 + b2 = c2	Formula (relating a, b and c)
x/2 + 7 = 0	Not a Formula (just an equation)

Without the Equals

Sometimes a formula is written without the "=":

Example: The formula for the volume of a box is:

lwh

But in a way the "=" is still there, because we can write V = lwh if we want to.

Subject of a Formula

The "subject" of a formula is the single variable (usually on the left of the "=") that everything else is equal to.

Example: in the formula

s = ut + ½ at²

"s" is the subject of the formula

Changing the Subject

One of the very powerful things that Algebra can do is to "rearrange" a formula so that another variable is the subject.

Rearrange the volume of a box formula (V = lwh) so that the width is the subject:

Start with:	V = lwh
divide both sides by h:	V/h = lw
divide both sides by l:	V/(hl) = w
swap sides:	w = V/(hl)

So now when you want a box with a volume of 12m³, a length of 2m, and a height of 2m, you can calculate its width:

w = V/(hl)

w = 12m³ / (2m × 2m) = 12/4 = 3m

Multiplying Negatives

When We Multiply:

			Example
×	two positives make a positive:		3 × 2 = 6
×	two negatives make a positive:		(−3) × (−2) = 6
×	a negative and a positive make a negative:		(−3) × 2 = −6
×	a positive and a negative make a negative:		3 × (−2) = −6

Yes indeed, two negatives make a positive, and we will explain why, with examples!

Signs

Let's talk about signs.

"+" is the positive sign, "−" is the negative sign.

When a number has no sign it usually means that it is positive.

Example: 5 is really +5

And we can put () around the numbers to avoid confusion.

Example: 3 × −2 can be written as 3 × (−2)

Two Signs: The Rules

"Two like signs make a positive sign, two unlike signs make a negative sign"

Example: (−2) × (+5)

The signs are − and + (a negative sign and a positive sign), so they are unlike signs(they are different to each other)

So the result must be negative:

(−2) × (+5) = -10

Example: (−4) × (−3)

The signs are − and − (they are both negative signs), so they are like signs (like each other)

So the result must be positive:

(−4) × (−3) = +12

Why does multiplying two negative numbers make a positive?

Well, first there is the "common sense" explanation:

When I say "Eat!" I am encouraging you to eat (positive)

But when I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

So, two negatives make a positive, and if that satisfies you, then you don't need to read any more.

Direction

It is all about direction. Remember the Number Line?

Well here we have Baby Steven taking his first steps. He takes 2 paces at a time, and does this three times, so he moves 2 steps x 3 = 6 steps forward:

Now, Baby Steven can also step backwards (he is a clever little guy). His Dad puts him back at the start and then Steven steps backwards 2 steps, and does this three times:

Once again Steven's Dad puts him back at the start, but facing the other way. Steven takes 2 steps forward (for him!) but he is heading in the negative direction. He does this 3 times:

Back at the start again (thanks Dad!), still facing in the negative direction, he tries his backwards walking, once again taking two steps at a time, and he does this three times:

So, by walking backwards, while facing in the negative direction, he moves in the positive direction.

Try it yourself! Try walking forwards and backwards, then again but facing the other direction.

More Examples

Example: Money

Imagine you owe Sam money.

Then Sam takes $10 of that debt away from you 3 times ... the same as giving you $30.

That is −$10 ($10 of debt) taken away 3 times (−3):

−$10 × −3 = +$30

Example: Tank Levels Rising/Falling

The tank has 30,000 liters, and 1,000 liters are taken out every day. What was the amount of water in the tank 3 days ago?

We know the amount of water in the tank changes by −1,000 every day, and we need to subtract that 3 times (to go back 3 days), so the change is:

−3 × −1,000 = +3,000

The full calculation is:

30,000 + (−3 × −1,000) = 30,000 + 3,000 = 33,000

So 3 days ago there were 33,000 liters of water in the tank.

Multiplication Table

Here is another way of looking at it.

Start with the multiplication table (just up to 4×4 will do):

×	1	2	3	4
1	1	2	3	4
2	2	4	6	8
3	3	6	9	12
4	4	8	12	16

Now see what happens when we head into negatives!

Let's go backwards through zero:

×	1	2	3	4
-4	-4	-8	-12	-16
-3	-3	-6	-9	-12
-2	-2	-4	-6	-8
-1	-1	-2	-3	-4
0	0	0	0	0
1	1	2	3	4
2	2	4	6	8
3	3	6	9	12
4	4	8	12	16

Look at the "4" column: it goes -16, -12, -8, -4, 0, 4, 8, 12, 16. Getting 4 larger each time.

Look over that table again, make sure you are comfortable with how it works, because ...

... now we go further to the left, through zero:

×	-4	-3	-2	-1	0	1	2	3	4
-4	16	12	8	4	0	-4	-8	-12	-16
-3	12	9	6	3	0	-3	-6	-9	-12
-2	8	6	4	2	0	-2	-4	-6	-8
-1	4	3	2	1	0	-1	-2	-3	-4
0	0	0	0	0	0	0	0	0	0
1	-4	-3	-2	-1	0	1	2	3	4
2	-8	-6	-4	-2	0	2	4	6	8
3	-12	-9	-6	-3	0	3	6	9	12
4	-16	-12	-8	-4	0	4	8	12	16

Same pattern: we can follow along a row (or column) and the values change consistently:

• Follow the "4" row along: it goes -16, -12, -8, -4, 0, 4, 8, 12, 16. Getting 4 larger each time.
• Follow the "-4" row along: it goes 16, 12, 8, 4, 0, -4, -8, -12, -16. Getting 4 smaller each time.
• etc...

So it all follows a neat pattern!

What About Multiplying 3 or More Numbers Together?

Multiply two at a time and follow the rules.

Example: What is (−2) × (−3) × (−4) ?

First multiply (−2) × (−3). Two like signs make a positive sign, so:

(−2) × (−3) = +6

Next multiply +6 × (−4). Two unlike signs make a negative sign, so:

+6 × (−4) = -24

Result: (−2) × (−3) × (−4) = −24