Solving Multi-Step Inequalities
1.  Solve
3x + 2 ≤ -7
x ≤ 3
x ≤ -3
x ≥ 3
x ≥ -3
Now, let's talk about interval notation...
x > 2 can be written as (2, ∞)
2
it's like the 
open dot
because 2 is
not a solution
infinity always
is paired with 
parentheses 
because the
infinity is not a #
Remember, when you divide both sides by a negative
number or multiply both sides by the same negative
number, you must flip the sign!
<    becomes    >
>    becomes    <
≤   becomes    ≥
≥   becomes    ≤
2.  Solve   6 - x > 8
x < -2
x < 2
x > -2
x > 2
In interval notation,
the solution is
(-2, ∞)
(-∞, -2)
(2, ∞)
(-∞, 2)
3.  Solve
2(x - 3) > -4
x > 5
x > -5
x > -1
x > 1
Interval notation
(-∞, 5)
(-∞, 1)
(1, ∞)
(-1, ∞)
4.  Solve  -3x + 7x > 20
x > 2
x > 5
x < -5
x > -5
(2, ∞)
(5, ∞)
(-∞, -5)
(-5, ∞)
In interval notation
5.  Solve  4 - 2(x - 1) < -8
x < -3
x > 7
x > -5
x < 7
(-∞, -3)
(-∞, 7)
(7, ∞)
(-5, ∞)
In interval notation
6.  Solve-x - 9 < -5
x < 4
x < -4
x > 4
x > -4
(-∞, 4)
(-∞, -4)
(-4, ∞)
(-∞, ∞)
In interval notation
Negative infinity is always on the left
Positive infinity is always on the right
In interval notation, (-∞, ∞) means...
All real numbers!
(-∞, 3)
(5, ∞)
7.  Solve6x + 2 > 4x - 8
x > 5
x > -5
x > 3
x > -3
In interval notation, if the solution includes (is equal to)
a number, you must use brackets. 
it's like the closed dot on the graph
[3, 5]
The solution includes 3 
and 5
- ∞
Left
Remember that you must read a graph from left to right
even though the arrows make it look like we start 
somewhere in the middle and read in both directions.
Infinity always gets parentheses!
to
right
In interval notation, this would look like
-5
[-5, 2]
-5 ≤ x ≤ 2
0
I think that interval notation
is easier; what do you 
think?
2
(set notation)
8.  Solve   x + 3 ≤ -7 - 4x
x ≤ 1
x ≥ 2
x ≤ 2
x ≤ -2
[1, ∞)
(-∞, -2]
[1, 2]
(-∞, 2]
Interval Notation
9.  Solve2(x + 11) > -3(x + 1)
x > -5
x < 5
x < -5
x > 5
10.  Solve -3  +  x  ≥  4    
x ≥ 2
x ≤ -14
x ≥ 14
x ≥ -12
2
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