This property is another fairly obvious one for real numbers – if Jack is shorter than Jill
and Jill is shorter than John then Jack is shorter than John!
- Ifa<bthena+c<b+c
Again fairly obvious – Jack doesn’t become any taller than Jill if they both stand on
the same chair.
- Ifa
0thenac < bc
This just says that if you multiply two numbersa,bby apositivenumbercthen the order
relation is unchanged – if you look at Jack and Jill through the same telescope, Jack is
still shorter than Jill.
- Ifa<bandc<0thenac > bc
This is usually where our troubles start with inequalities, when we bring in negative
multipliers. This property says that changing the sign throughout reverses the inequality.
It is pretty obvious if you just think of an example: 4>3, but− 4 <−3and− 8 <−6,
and plottinga,b,−aand−bon the real line may make it clear generally. Jack and Jill
can help by hanging from the ceiling – Jack is now ‘higher than Jill’! More rigorously,
note that we can writeb=a+pwherepis a positive number and so ifcis a negative
number thenbc=ac+pc,soac=bc−pc.Butifpis positive andcis negative,−pc
is positive, from which it follows thatac > bc.
- Ifa
0then
1
a>1
bOur troubles are mounting now – reciprocals and inequalities cause problems for
most of us. Again it helps to think of numbers: 2<3, and^12 >^13 butwhile− 2 <3,
1
− 2 =−
1
2 <1
3. So the difference in sign affects the inequality when we take reciprocals.
The condition ab >0 is included to make sure that a and b have the same sign,
both positive, or both negative. Only then does the inequality reverse if we take the
reciprocals. The contortions that Jack and Jill have to get up to to convince you of this
don’t bear thinking about. Perhaps it’s best if you simply satisfy yourself with a few
examples!
Most of the above results may be extended, with care, to≤and≥.
Inequalities can be used to specifyintervalsof numbers, for which a special notation
is sometimes used. Specifically, [a,b] denotes theclosedinterval a≤x≤band (a,b)
denotes theopenintervala<x<b.
Sometimes we have a problem in which we need to find values ofxsuch that a given
functionf(x)satisfies an inequality such as
f(x)< 0We call thissolving the inequality. In such problems we may have to combine algebra,
knowledge of the function, and properties of inequalities. It is a common error to try to
solve such an inequality by considering theequalityf(x)=0 instead. Whilst doing this
may be useful, remember that it will only give theboundariesof the inequality region,
and extra work is needed to define the entire region.