1. Inequalities
An inequality such as 4x - 6 < 10 is similar to the equation 4x - 6 = 10. The difference is we are
looking for numbers which if you multiply by 4, then subtract 6, you get an answer of less than 10,
and unlike 4x - 6 = 10, whichhas only one answer,there are lotsof numbersforwhichthisis true. So
our answer is not a number, but a range of numbers. Inequalities are solved just like simple
equations, what you do to one side, you must do to the other.
There are four main symbols used when dealing with inequalities:
< is less than
> is more than
is less than or equal to
is more than or equal to
Example question and answer:
Question
Solve the expression 4x - 6 < 10
Answer
The first step is to write down the inequality.
4x – 6 < 10
The next step is to add 6 to both sides, this cancelling the -6.
4x < 16
Now simplify the inequality by dividing both side by 4.
x < 4
Therefore the inequality 4x – 6 < 10 is satisfied when x is any number less than 5.
You may also be given a simultaneous inequality to solve, the same rules apply for these types of
questions however there are three parts appose to two.
Example question and answer:
Question
Solve the inequality -6 < 4x – 8 < 16
Answer
The first step is to write down the inequality.
-6 < 4x – 8 < 16
The next step is to add 8 to all parts.
2. 2 < 4x < 24
Now simplify the inequality by dividing both side by 4.
0.5 < x < 6
Therefore the inequality -6 < 4x – 8 < 16 is satisfied when x is more than 0.5 but less than 6.
Another type of inequality that may be given is a quadratic inequality. With quadratic inequalities
youmust getall the unitstoone side makingthemequal zeroandthenfactorise.Once factorised,by
plotting a graph it will help make it easier to see what values of x are true for the inequality.
Example question and answer:
Question
Solve the inequality x2
- 9x < 36
Answer
The first step is to write down the inequality.
x2
- 9x < 36
The next step is to get all the units to one side and make them equal to 0.
x2
- 9x - 36 = 0
Now factorise the quadratic.
(x - 12)(x + 3) = 0
Next calculate the critical values of (x - 12)(x + 3) = 0 to find x = 12 or x = -3.
Now sketch the graph and mark the x coordinates of intersection. Then highlight the
graph where the x values are less than 0.
By looking at the graph and the factorised inequality you can see x > -3 but x < 12,
therefore giving the answer -3 < x < 12.
