The number line

On the number line,  a < b  means:  a falls to the left of b.

Problem 1.   Between each pair, place the correct sign of inequality.

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Do the problem yourself first!

"Or" versus "and"

The following is called a compound inequality:

x > 1  and   x ≤ 5.

It is a compound sentence whose conjunction is "and."  It says that x takes on values that are greater than 1  and  less than or equal to 5.

It is within that interval that x takes its values.  The endpoint 1 is not included.  x is definitely greater than 1; we indicate that by placing a parenthesis "(".  The endpoint 5 is included; we indicate that by placing a bracket "]".

Now consider this compound inequaltiy:

x < 1  or  x > 5.

Here, the values of x are either less than 1  or  greater than 5.

It should be clear that x could not be a number that is less than 1 and greater than 5.  There is no such number.

So, when the conjunction is and,

x > 1  and   x ≤ 5.

the values of x fall inside a certain interval.  But when the conjunction is or,

x < 1  or  x > 5.

the values of x fall outside an interval.

Problem 2.   Graph these compound inequalities.

a)     x < −1   or   x ≥ 3.

b)     x ≥ −1   and   x < 3.

A continued inequality

a < x < b

That is called a continued inequality.  It means

a < x  and  x < b.

A continued inequality always implies the conjunction and.  The sense is always <  or  ≤.

The continued inequality means:

x falls in the interval between a and b.

"a < x < b" illustrates that.

Note:  When neither end point is included, as in this example, we call that an open interval.  When both end points are included --

-- we call that a closed interval.  (Otherwise, like the proverbial glass, the interval is half-open, or half-closed.)

Problem 3.   Write as a continued inequality.

a)   x > −3  and  x < 1.

−3 < x < 1.

b)  Graph that continued inequality.

Problem 4.   0 < x < 6.  Write that continued inequality as a compound inequality.

x > 0  and  x < 6.

Problem 5.   Write as a continued inequality:

x < 1  or  x > 5.

Not possible!  The conjunction must be and.

Problem 6.   Name four values that x might have.

a)   1 ≤ x < 3.

For example, 1,  1.2,  2.5,  2.999999999999999999999.

b)  x < −1,  or   x > 1.

For example, −2,  −3,456,987,  1.000005,  10²³.

Problem 7.   Write in symbols.

a)  x is a positive number.   x > 0

b)  x is a negative number.   x < 0

c)  x is a non-negative.   x ≥ 0.  "x is greater than or equal to 0."

Some theorems of inequalities

Algebraically, we say that  a > b  when  a − b is positive.

Theorem 1.    We may add the same number to both sides of an inequality, and the sense will not change.

We can prove the theorem as follows:

Which is what we wanted to prove.

Theorem 2.   We may multiply both sides of an inequality by the same positive number, and the sense will not change.

For example:

The sense does not change.

Theorem 3.   If we multiply both sides of an inequality by the same negative number, the sense of the inequality changes.

Here is an example:

The sense changes.

Proof:

Theorem 4.   If we change the signs on both sides of an inequality,
then the sense of the inequaltiy will change.

We can see that on the number line.

Theorem 5.   If a, b are both positive or both negative, then on taking reciprocals, the sense of the inequality changes.

Problem 11.   −a > 0.  How is that possible?

a is not a number. It is a variable. It takes values which are numbers. In this case, a must have a negative value; for example, −5. Therefore, −a = −(−5)  is a positive number.

Solving inequalities

A linear inequality has this standard form:

ax + b < c