3.4 Solving Multi-Step inequalities:
Solution of an Inequality: Any number that
makes the inequality true.
Inverse Operations: Operations that undo
another operation.
Isolate: The use of inverse operations
used to leave a variable by itself.
GOAL:
Multi-Step inequalities:
Just like the equations, we can isolate a
variable in an inequality by using
inverse operations.
Ex:
What are the solutions to
9 + 4t > 21
Check the solutions.
http://www.khanacademy.org/math/algebra/linear_ineq
ualities/inequalities/v/multi-step-inequalities
SOLUTION:
9 + 4t > 21
Given
4t > 21 - 9
Inverse of addition
4t > 12
𝟒𝒕
𝟒
>
𝟏𝟐
𝟒
t >3
Inverse of multiplication
 interval: (3, ∞)
CHECK:
9 + 4t > 21
9+ 4(4) > 21
9 + 16 > 21
t > 3,
we cannot use 3
but 4 and on…
25 > 21
25 > 21  TRUE
Remember: sign switches direction when
dividing or multiplying by a negative number.
YOU TRY IT:
What are the solutions
to
3( t + 1) – 4t ≥ -1?
http://www.khanacademy.org/math/algebra/
linear_inequalities/inequalities/v/multi-stepinequalities-2
SOLUTION:
3( t + 1) – 4t ≥ -1
3t + 3 - 4t ≥ -1
3 - t ≥ -1
Given
Distributive property
Add like terms
- t ≥ -1 -3
Inverse of addition
-t≥-4
t≤ 4
(-∞,4]
Add like terms
÷ by -1, Sign switches
Interval
Real-World:
You have taken a quiz and got 45%. You
are about to take another quiz next
week. If you want to pass the portion of
quizzes in the class you must get an
average of at least 70%. What is the
minimum percentage you must get on
the next quiz?
Real-World: (SOLUTION)
Quiz 1 = 45%
Quiz 2 = x%
Average 
𝑸𝒖𝒊𝒛 𝟏+𝑸𝒖𝒊𝒛 𝟐
𝟐
At least  ≥
𝟒𝟓% +𝑿
𝟐
≥ 𝟕𝟎%
45% + x ≥ 𝟏𝟒𝟎%
x ≥ 𝟏𝟒𝟎% − 𝟒𝟓%
Thus in order for you to get an average of 70%
in your quizzes, you must get at least 95% on
the second quiz.
Real-World:
In a community garden, you want to
plant and fence in a vegetable garden
that is adjacent to your friend’s garden.
You have at most 42 ft. of fence. What
are the possible lengths of your garden?
Since the fence will go
around the garden, we use:
SOLUTION:
Perimeter = 𝟐𝒍 + 𝟐𝒘
𝟐𝒍 + 𝟐𝒘 ≤ 42 At most 42 ft. of fence
𝟐𝒍 + 𝟐(𝟏𝟐) ≤ 42 Width on picture = 12 ft
𝑻𝒉𝒆 𝒍𝒆𝒏𝒈𝒉𝒕
𝟐𝒍 + 𝟐𝟒 ≤ 42
𝒘𝒐𝒖𝒍𝒅
𝟐𝒍 ≤ 42 - 24
𝒖𝒑 𝒕𝒐 𝒃𝒖𝒕
𝟐𝒍 ≤ 18
𝒏𝒐 𝒎𝒐𝒓𝒆𝒕𝒉𝒂𝒏 𝟗 𝒇𝒕.
𝒍 ≤ 9
WRITING SET AND INTERVAL NOTATION:
Always isolate variable, graph and write
interval:
x≤0
Set Notation: { x| x ≤ 0}
Interval Notation: (-∞, 0]
x ≥ 2
Set Notation: { x| x ≥ 2}
Interval Notation: [2, ∞)
WRITING SET AND INTERVAL NOTATION:
Always isolate variable, graph and write
interval:
x < -1
Set Notation: { x| x < -1}
Interval Notation: (-∞, -1)
x>1
Set Notation: { x| x > 1}
Interval Notation: (1,∞)
VIDEOS:
Inequalities
https://www.khanacademy.org/math/algebra/line
ar_inequalities/inequalities/v/multi-stepinequalities
https://www.khanacademy.org/math/algebra/line
ar_inequalities/inequalities/v/multi-stepinequalities-2
CLASSWORK:
Page 189-192
Problems: As many as needed
to master the
concept.