© Daniel Holloway
3 > -6 is an inequality. It is also a true
statement, so 3 > -6 is a true inequality.
But does the inequality remain true when
you do the following to it?
add 4 to both sides:
3 + 4 > -6 + 4
7 > -2
is still true
subtract 2 from both sides:
3 – 2 > -6 – 2
1 > -8
is still true
multiply both sides by 2:
3 x 2 > -6 x 2
6 > -12
is still true
divide both sides by 3:
3 ÷ 3 > -6 ÷ 3
1 > -2
is still true
multiply both sides by -5:
3 x -5 > -6 x -5
-15 > 30
is not true
divide both sides by -3:
3 ÷ -3 > -6 ÷ -3
-1 > 2
is not true
This means that to solve an inequality, you
can:
 add the same quantity to both sides
 subtract the same quantity from both sides
 multiply both sides by the same positive
quantity
 divide both sides by the same positive
quantity
But you cannot:
 multiply both sides by a negative quantity
 divide both sides by a negative quantity
 Worked Example
a)
Solve 3x + 4 > 22
b)
Solve 6 – 5x ≥ 3x + 2
a
subtract 4 from both sides
divide both sides by 3
3x + 4 – 4 > 22 – 4
3x ÷ 3 > 18 ÷ 3
b
add 5x to both sides
6 – 5x + 5x ≥ 3x + 5x + 2
6 ≥ 8x + 2
6 – 2 ≥ 8x + 2 – 2
4 ≥ 8x
4 ÷ 8 ≥ 8x ÷ 8
½≥x
subtract 2 from both sides
divide each side by 8
Another way of writing ½ ≥ x is x ≤ ½
3x > 18
x>6
Solve these inequalities:
1) 3x – 5 ≥ 4
5)
2) 5x + 2 > 4
6)
3) 7x + 2 < 5x + 4
7)
4) 8 – 3a ≤ 5
8)
-2d ≥ -3
-5c < 20
5 – ¼f > 12
2 – ½g ≥ 1
We have to treat inequalities with three sides as
two separate two-sided ones
 Worked Example
Solve 7 ≤ 3x – 2 < 9
Treat this as two inequalities:
7 ≤ 3x – 2
3x – 2 < 9
9 ≤ 3x
3x < 11
3≤x
x < 32 /3
We write the answer as:
3 ≤ x < 32 /3
Inequalities can be plotted on a graph
using regions (R)
y
R
5
0
5
x
The shaded region
here represents the
inequality. This
region satisfies the
inequality:
x>3
The dotted line
means it is “more
than but not
including” or >
y
The shaded region
in this grid shows
the inequality:
5
y≤6
The filled in line
means it is “below
and including” or ≤
R
0
5
x
We can use multiple regions on the same
graph to find points which satisfy several
inequalities at the same time
For example, on the following grid, if we
wanted to find the region which satisfied
the three inequalities:
y<7
x>1
y≥x
As you can see, all
three inequalities
have been drawn
on the grid, and this
time, those areas
which do not fit are
shaded (or crossed
out) in order to
leave the last region
y
5
0
5
x
y<7
x>1
y≥x