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

How to Solve

Solving inequalities is very like solving equations... you do most of the same things ...

... but you must also pay attention to the direction of the inequality.

Direction: Which way the arrow "points"

Some things you do will change the direction! When you multiply or divide by a negative number you have to reverse the inequality.

< would become >

> would become <

≤ would become ≥

≥ would become ≤

Safe Things To Do

These are things you 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

You 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 you swap the left and right hand sides, you 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 you 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 .

But we need to be a bit more careful (as you will see).

Positive Values

Everything is fine if you 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 you multiply or divide by a negative number
you have to 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 you 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 you 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 could you solve something where there are two inequalities at once?

Solve:

-2 < (6-2x)/3 < 4

First, let us clear out the "/3" by multiplying each part by 3:

Because you 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 you 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)

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