Inequalities
Symbols
Here are all the symbols used in inequalities which you will come across:
a<b
a≤b
a≥b
a>b
means a is less than b (so b is greater than a)
means a is less than or equal to b (so b is greater than or equal to a)
means a is greater than or equal to b etc.
means a is greater than b etc.
Solving inequalities
When you are given an inequality you have to treat it like it were a normal equation but with the = sign
replaced with an inequality. You would then solve the inequality as if it were any other equation. For
example:
Remember: If you divide with a negative number you have to flip the
inequality sign. E.g. -3x < 12 would become x > -4
2x + 7 < 11
Also remember: If you are given an inequality similar to this 5 < 7 – 2x
< 13, just split in two e.g. 5 < 7 – 2x and 7 – 2x < 13. Once you have
worked it out put the smallest number on the left and join the two
answers together again e.g. -3 < x < 1
2x < 4
X<2
Quadratic Inequalities
If you have to solve a quadratic inequality first of all you have to
draw the graph.
This involves solving the quadratic to work out the two values of x and the point of intersection
on the y axis.
Once you have done this you then look at the different parts of the graph depending on the
inequality.
So for example if f(x) is greater than 0 you look at the top part of the graph and if f(x) is less than 0
you look at the lower part of the graph. From this you can solve the inequality.
Examples: x^2-6x+8>0
Because f(x) is greater than 0 you look
at the top part of the graph.
Therefore, here x < 2
Always label
the points on
your graph
Therefore, here x > 4
This means the answer is: x<2, x>4
If it had been x^2-6x+8<0 then you would have had to look at the lower part of the graph and the
answer would have been 2 < x < 4
Inequalities
Notation for inequalities
When comparing the sizes of numbers using the inequality symbols can help illustrate whether or
not one number is bigger than the other. These symbols being:




Greater than, >
Less than, <
Greater than or equal to, ≥
Less than or equal to, ≤
So when put into action: ‘y < x’ tells us that y is less tan x. It doesn’t matter what the actual values of
y and x are or whether they are positive or negative as long as y in less than x then the equation is
correct. ‘y < x’ is the same as saying ‘x > y’.
Linear inequalities
When solving linear inequalities the goal is to simplify the inequality as much as possible, for
example:
2x + 3 < -5
2x < -8
x < -8/2
x < -4
When solving the inequality you are required to add, subtract, divide and multiply where
appropriate but bear in mind that when dividing or subtracting by a negative number the inequality
sign is flipped in the opposite direction, for example:
-2x < 8 when simplified becomes,
x > -4
Notice that the symbol is now facing the opposite direction because we have had to divide by
‘negative 2’ in order to solve the inequality.
Quadratic inequalities
How to solve the inequality ‘(x-3)(x-5) < 0’
Sketch graph of y=(x-3)(x-5)
Coefficient of x² is positive so parabola
bends upwards and it intercepts the x axis
at 3 and 5.
From the graph you can see that that y > 0
when 3 < x < 5 so x is greater than 3 but less
than 5.