For examples, consider the possible inequalities
relating the expressions 2*x* − 4 and 3*x* − 9.
Let's evaluate these five inequalities,
each at *x* = 4 and each at *x* = 5.
As before, when *x* = 4,
then 2*x* − 4 = 4 and 3*x* − 9 = 3;
when *x* = 5,
then 2*x* − 4 = 6 and 3*x* − 9 = 6.
So here is a table of the results:

Inequality: | When x = 4: |
Result: | When x = 5: | Result: |
---|---|---|---|---|

2x − 4 < 3x − 9; |
4 < 3; | False; | 6 < 6; | False. |

2x − 4 > 3x − 9; |
4 > 3; | True; | 6 > 6; | False. |

2x − 4 ≤ 3x − 9; |
4 ≤ 3; | False; | 6 ≤ 6; | True. |

2x − 4 ≥ 3x − 9; |
4 ≥ 3; | True; | 6 ≥ 6; | True. |

2x − 4 ≠ 3x − 9; |
4 ≠ 3; | True; | 6 ≠ 6; | False. |

t= 1 ort= −1.

A **compound statement**
consists of simpler statements (like equations or inequalities)
joined by either the word ‘and’ or the word ‘or’.
(If you study Logic,
you'll see that there are other words that might be used here,
like ‘if’ or ‘but not’.
However, we won't need such connectives in this course.)
For example:

*t*= 1 or*t*= −1;*x*= 2 or*x*= 3;*x*= 2 or*y*= 3;*x*< 2 or*x*≥ 3;*x*> 2 and*x*≤ 3;*p*= 1 or*p*= 5 or*p*= 7;*x*> 2 and*x*≤ 3, or*x*> 4.

Particularly common are **compound inequalities**,
each of which consists of two inequalities joined with ‘and’
and sharing one expression.
An example is

above. This example can be summarised simply asx> 2 andx≤ 3

2 <that is its form as a compound inequality. (Notice thatx≤ 3;

Here are some more examples:

Compound inequality: | Expanded meaning: |
---|---|

2 < x ≤ 3; |
2 < x and x ≤ 3; |

−1 < y < 6; |
−1 < y and y < 6; |

5 > a ≥ 3; |
5 > a and a ≥ 3; |

2 < 5x − 9 ≤ 3; |
2 < 5x − 9 and
5x − 9 ≤ 3. |

The words ‘and’ and ‘or’
are pretty fundamental English words,
but for purposes of Mathematics I should tell you exactly what they mean.
A compound statement with ‘and’
is true only if *both* statements are true,
and false if *either* statement (or both!) is false.
Conversely, a compound statement with ‘or’
is true if *either* statement (or both!) is true,
and false only if *both* statements are false.
Finally, if any statement within a compound statement is meaningless,
then so is the compound statement.

In particular, mathematicians
always use ‘or’ in an *inclusive* sense,
so that if both statements are true, then the compound statement is true.
For example, consider the compond statement
*x* ≤ 2 or *x* < 3.
This is really just equivalent to the simple statement *x* < 3,
because both parts are true when *x* ≤ 2,
so the first part (*x* ≤ 2) is superflouous.

To begin with, consider the equation

It's easy to draw a picture of this on a number line:x= 4.

This is so easy that there may not seem much point to it. However, there are a few points worth mentioning:

- The line has arrows at either end,
because
*it*goes on forever, even though the solution is at only one place. - A scale is marked on the line aside from just the point 4, so you can see the solution in context.
- The line is labelled with the variable
*x*, so you know that the graph is for*x*= 4, rather than*y*= 4 or*t*= 4.

Now consider a more complicated solution, the compound statement

Now the graph looks a little more interesting:x= −2 orx= 4 orx= 6.

You can see here what I mean when I say that the parts of a compound statement should be listed in increasing order;

means the same thing, but it doesn't match the graph as nicely.x= 4 orx= 6 orx= −2,

Now consider the inequality

Here,y< 3.

Since

Compare this with

y≥ 3:

Now I have a square bracket instead of a round parenthesis, to indicate that now

There is another way to graph inequalities, which you may have already learned before this course. I think that it's less clear than the method above, but you can use it if you like. In this method, you use a solid dot (instead of a square bracket) to indicate that a boundary point is a solution, and you use a hollow circle (instead of a round parenthesis) to indicate that a boundary point is not a solution. Then you get these graphs:

If you use the circle/dot method, then be sure to shade in well the region where the solutions are; this isn't so important when you use the parenthesis/bracket method.

Now consider the compound inequality

−3 <This means that −3 <x≤ 2.

You could just as well write this compound inequality as 2 ≥

Finally, consider the compound statement

This comes in two pieces, one wherex≤ −3 orx> 2.

Again, you could just as well write

Notice that these two pieces have no overlap. Compare this with a statement like

This statement isx≤ 3 orx> −2.

True.That has a very simple graph;

The flip side of this idea is a compound inequality like

2 <Because 2 > −3, no real number can possibly be both greater than 2x≤ −3.

False.Its graph is the simplest of all, completely empty:

- If there is a limited range of values,
as we have with a compound inequality,
give the first value and the last value, separated by a comma;
put round parentheses or square brackets around this pair,
depending on whether these values are or are not included.
For example, the solution set for
*x*of −3 <*x*≤ 2 is (−3, 2]; the round parenthesis around −3 indicates that −3 is not included, while the square bracket around 2 indicates that 2 is included. - If there is an unlimited range of values in the negative direction,
then use the symbol ‘−∞’
(pronounced ‘minus infinity’)
as the first value;
always use a round parenthesis there
(since −∞ itself is not a real number,
so it can't possibly be a solution).
For example, the solution set for
*y*of*y*< 3 is (−∞, 3), while the solution set for*y*of*y*≤ 3 is (−∞, 3]. - If there is an unlimited range of values in the positive direction,
then use the symbol ‘∞’ (pronounced ‘infinity’)
as the last value;
again always use a round parenthesis there.
For example, the solution set for
*y*of*y*> 3 is (3, ∞), while the solution set for*y*of*y*≥ 3 is [3, ∞). - If there are two or more disjoint ranges of values,
then list them all,
separated by the symbol ‘∪’ (pronounced ‘union’);
again, it's helpful to list these in increasing order.
For example, the solution set for
*x*of*x*≤ −3 or*x*> 2 is (−∞, −3] ∪ (2, ∞).

These continuous ranges (as I've been calling them)
are **intervals** in the real line;
writing solutions sets this way is called *interval notation*.
(The first couple of examples aren't really interval notation.)

Compare the interval notation to the graphs; you'll see that (except for −∞ and ∞) the round parentheses and square brackets match up perfectly; they're used in the same places, facing in the same directions. This is no coincidence, of course; the notation is designed to work this way!

To solve an *order* inequality, in contrast,
is a little trickier.
You can basically use the same techniques as for solving equations,
but there are complications.
The first technique
—replacing one expression by an equivalent expression—
is exactly the same.
The last technique —swapping the sides— is almost the same,
but you also have to switch the direction on the inequality.
(For example, 2 < *x* is equivalent to *x* > 2.)
But the main technique
—doing the same invertible operation to each side—
is more complicated;
the operation must *also* preserve order.

An **order-preserving** operation
is any operation on a real number
that always takes smaller numbers to smaller numbers
and larger numbers to larger numbers.
Notice that this is *relative*;
adding a million may take small numbers to large numbers,
but as long as one number is small*er* than another,
then it will still be smaller after you add a million to both of them.

Addition and subtraction always preserve order
(as long as you are subtracting *from* the two sides of the inequality).
Multiplication or division by a *positive* quantity
is also order-preserving.
But multiplication or division by a *negative* quantity
does *not* preserve order!
For example, start with 2 and 3; notice that 2 < 3.
After you multiply these by the negative number −4,
you get (−4)2 = −8 and (−4)3 = −12,
but −8 > −12.

In fact, multiplication or division by a negative quantity
*reverses* order.
An **order-reversing operation**
is any operation on a real number
that always takes smaller numbers to *larger* numbers
and larger numbers to *smaller* numbers.
You can still use such an operation to solve order inequalities,
but you must switch the direction of the inequality when you do so!

So here's a summary of techniques for solving order inequalities:

- Replace either side (or both) with an equivalent expression (usually simplified);
- Add the same expression to both sides (and then simplify the results);
- Subtract the same expression from both sides (and simplify);
- Multiply both sides by the same positive expression (and simplify);
- Divide both sides by the same positive expression (and simplify);
- Multiply both sides by the same negative expression (and simplify)
*and*switch the direction of the inequality; - Divide both sides by the same negative expression (and simplify)
*and*switch direction; - Swap the sides
*and*switch direction.

- 5
*y*+ 4 < 2*y*− 2 — original inequality; - 3
*y*+ 4 < −2 — subtract 2*y*from both sides; - 3
*y*< −6 — subtract 4 from both sides; *y*< −2 — divide both sides by 3.

Here's an example where I have to change the direction of the inequality:

- 2
*c*+ 5 ≥ 3*c*+ 7 — original inequality; - −
*c*+ 5 ≥ 7 — subtract 3*c*from both sides; - −
*c*≥ 2 — subtract 5 from both sides; *c*≤ −2 — take the opposite of both sides.

To solve compound statements, you can just treat each statement separately. However, if you have a compound inequality, like

2 <then instead of breaking this up into 2 <x+ 1 ≤ 5,

1 <that's the answer. To check whenx≤ 4;

Go back to the course homepage.

This web page was written in 2007 and 2008 by Toby Bartels. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-0950/2008WN/inequalities/`

.