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Inequalities &
Integers
Learn to solve inequalities with integers.
Remember!
The graph of an inequality shows all of
the numbers that satisfy the inequality.
When graphing inequalities on a number
line, use solid circles ( ) for  and 
and open circles ( ) for > and <.
Solve and graph.
A. k +3 > –2
k +3 > –2
–3
–3
k > –5
–5
Subtract 3 from both sides.
0
Solve and graph.
B. r – 9  12
r – 9  12
r – 9 + 9  12 + 9
Add 9 to both sides.
r  21
15
21
24
Sometimes you must multiply or divide to isolate
the variable. Multiplying or dividing both sides of
an inequality by a negative number gives a
surprising result.
5 is greater than –1.
5 > –1
–1 • 5 –1 • (–1) Multiply both sides by –1.
> or < ?
–5
1
–5 < 1
You know –5 is less than 1, so you should use <.
–5 < 1
–7 –6 –5 –4 –3 –2 –1
0
1
2 3 4 5 6 7
5 > –1
MULTIPLYING INEQUALITIES BY
NEGATIVE INTEGERS
Words
Multiplying or
dividing by a
negative
number
reverses the
inequality
symbol.
Original
Inequality
Multiply/
Divide
Result
3>1
Multiply by –2
–6 < –2
–4  12
Divide by –4
1  –3
Helpful Hint
The direction of the inequality changes
only if the number you are using to
multiply or divide by is negative.
Solve and graph.
A. –3y  15
Divide each side by
–3;  changes to .
–3y  15
–3
–3
–7 –5
0
y  –5
B. 7m < 21
7m < 21
7
7
m<3
Divide each side by 7.
–3
0
3
5
4
Solve and graph.
1. h + 2 < 0
h  –2
–2 0 2
2. c – 5 > –2
c3
3.
4.
t
–3
–3
0
3
–3
0
3
0
4
<1
t > –3
7n > 28
n>4
8
Two-step Inequalities.
Solve and graph.
1. -3 – 2n > 1
n  –2
–2 0 2
2. -4 – 2r < –6
0
r>1
3.
4.
x
5
–1>1
x  10
0
+5<6
6r < 6
0
r
10
6
9
Two-step Inequalities.
Solve and graph.
1. -1 + 2m > 15
n8
0
8
2. -5k + 1 > 21
k < -4
3.
4.
t
4
x
-4
0
-4
0
+5>4
t  -4
+1>0
8 x  -8
-8
0