Solving Inequalities

Sometimes we need to solve Inequalities like these:

Symbol

Words

Example

>

greater than

x + three > two
<

less than

7x < 28
≥

greater than or equal to

5 ≥ x − i

≤

less than or equal to

2y + i ≤ 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: ten < v
or: y ≥ 11

Nosotros call that "solved".

Example: ten + ii > 12

Subtract 2 from both sides:

x + 2 − 2 > 12 − two

Simplify:

x > 10

Solved!

How to Solve

Solving inequalities is very similar solving equations ... we do nearly of the aforementioned things ...

... but nosotros must too pay attention to the direction of the inequality.

greater than sign
Direction: Which way the arrow "points"

Some things can modify the direction!

< becomes >

> becomes <

≤ becomes ≥

≥ becomes ≤

Safe Things To Do

These things do not affect the management of the inequality:

  • Add (or decrease) a number from both sides
  • Multiply (or separate) both sides past a positive number
  • Simplify a side

Example: 3x < 7+3

Nosotros can simplify 7+three without affecting the inequality:

3x < 10

But these things do change the direction of the inequality ("<" becomes ">" for example):

  • Multiply (or separate) both sides past a negative number
  • Swapping left and right hand sides

Example: 2y+seven < 12

When we bandy the left and correct paw sides, we must also modify the direction of the inequality:

12 > 2y+seven

Here are the details:

Adding or Subtracting a Value

We tin can often solve inequalities by adding (or subtracting) a number from both sides (merely equally in Introduction to Algebra), like this:

Example: 10 + 3 < vii

If nosotros decrease 3 from both sides, nosotros go:

x + 3 − 3 < vii − three

x < four

And that is our solution: x < four

In other words, ten can exist any value less than 4.

What did we practice?

We went from this:

To this:

 number line inequality x+3 < 7

x+3 < 7

x < iv

And that works well for calculation and subtracting, considering if nosotros add (or subtract) the same amount from both sides, it does not affect the inequality

Example: Alex has more coins than Baton. If both Alex and Baton get three more coins each, Alex will still have more coins than Billy.

What If I Solve Information technology, But "x" Is On The Right?

No matter, but swap sides, but reverse the sign so it however "points at" the right value!

Case: 12 < 10 + 5

If we subtract 5 from both sides, we go:

12 − five < x + 5 − v

7 < x

That is a solution!

But it is normal to put "x" on the left hand side ...

... so allow united states flip sides (and the inequality sign!):

10 > seven

Practise you run into how the inequality sign notwithstanding "points at" the smaller value (7) ?

And that is our solution: 10 > 7

Note: "10" tin be on the right, but people usually like to come across it on the left mitt side.

Multiplying or Dividing by a Value

Some other thing nosotros do is multiply or carve up both sides by a value (just equally in Algebra - Multiplying).

Simply nosotros need to exist a bit more careful (as yous will encounter).


Positive Values

Everything is fine if we want to multiply or separate past a positive number:

Example: 3y < 15

If we carve up both sides by iii we get:

3y/three < 15/3

y < 5

And that is our solution: y < 5


Negative Values

warning! When we multiply or split by a negative number
we must contrary the inequality.

Why?

Well, but expect at the number line!

For instance, from 3 to 7 is an increment,
but from −iii to −7 is a decrease.

number line -vii<-three and 3<7
−vii < −3 vii > 3

Run across how the inequality sign reverses (from < to >) ?

Let usa effort an example:

Example: −2y < −viii

Let us divide both sides by −2 ... and contrary the inequality!

−2y < −8

−2y/−2 > −8/−ii

y > 4

And that is the right solution: y > 4

(Notation that I reversed the inequality on the same line I divided by the negative number.)

Then, just retrieve:

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

Multiplying or Dividing by Variables

Hither is some other (tricky!) case:

Example: bx < 3b

It seems easy just to carve up both sides by b, which gives u.s.a.:

x < 3

... but wait ... if b is negative we need to opposite the inequality like this:

x > 3

Only nosotros don't know if b is positive or negative, so nosotros can't answer this one!

To help y'all understand, imagine replacing b with 1 or −1 in the case of bx < 3b:

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

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

Then:

Do not try dividing by a variable to solve an inequality (unless you know the variable is e'er positive, or ever negative).

A Bigger Example

Example: x−iii ii < −5

First, let us clear out the "/2" by multiplying both sides past 2.

Because nosotros are multiplying past a positive number, the inequalities volition not modify.

x−three 2 ×2 < −5×two

x−3 < −10

Now add 3 to both sides:

x−3 + 3 < −x + three

ten < −7

And that is our solution: x < −seven

Ii Inequalities At Once!

How practise we solve something with two inequalities at once?

Example:

−2 < 6−2x 3 < 4

Get-go, allow us articulate out the "/3" by multiplying each part by 3.

Because we are multiplying by a positive number, the inequalities don't modify:

−half-dozen < 6−2x < 12

At present subtract 6 from each part:

−12 < −2x < 6

At present split each part by 2 (a positive number, and then again the inequalities don't change):

−half dozen < −x < 3

Now multiply each part past −i. Because we are multiplying by a negative number, the inequalities modify direction.

half dozen > ten > −3

And that is the solution!

But to be neat information technology is amend to have the smaller number on the left, larger on the correct. So let us swap them over (and make certain the inequalities point correctly):

−iii < x < vi

Summary

  • Many elementary inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
  • Only these things will alter 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)