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Solving Multi-Step Inequalities
Solve each inequality.
1.  9v  108
v | v  12
2. 2c  5
3
9
5

c | c   
6

3. Which inequality is not represented by the graph:
Solving Multi-Step Inequalities
Solving inequalities involving more than one
operation can be solved by undoing the
operations in the same way you would solve an
equation with more than one step.
Solve a Real-World Problem
Science The inequality F > 212 represents the
temperatures in degrees Fahrenheit for which water is
a gas (steam). Similarly, the inequality
represents the temperatures in degrees Celsius for
which water is a gas. Find the temperature in degrees
Celsius for which water is a gas.
Solve a Real-World Problem
Original inequality
Subtract 32 from each side.
Simplify.
Multiply each side by
Simplify.
Answer: Water will be a gas for all temperatures
greater than 100°C.
Do this one in your notes, please!
Solve a Real-World Problem
Science The boiling point of helium is –452°F. Solve
the inequality
to find the temperatures
in degrees Celsius for which helium is a gas.
9
C  484
Subtract 32 from each side
5
5
C  9  484
Multiply each side by 5/9
 

Answer: Helium will be a gas for all temperatures
greater than –268.9°C.
Inequality Involving a Negative Coefficient
Then check your solution.
Original inequality
Subtract 13 from each side.
Simplify.
Divide each side by –11 and
change
Simplify.
Inequality Involving a Negative Coefficient
Check To check the solution, substitute –6, a number
less than –6, and a number greater than –6.
Answer: The solution set is
Do this one in your notes, please!
Inequality Involving a Negative Coefficient
Then check your solution.
 8 y  8
y 1
Answer:
Subtract 3 from each side
Divide each side by – 8
AND reverse the inequality
Write and Solve an Inequality
Write an inequality for the sentence below. Then solve
the inequality.
Four times a number plus twelve is less than a number
minus three.
Let n = a number
Four times
a number
plus
twelve
is less
than
4n
+
12
<
a number
minus three.
Write and Solve an Inequality
Original inequality
Subtract n from each side.
Simplify.
Subtract 12 from each side.
Simplify.
Divide each side by 3.
Simplify.
Answer: The solution set is
Write and Solve an Inequality
Write an inequality for the sentence below. Then solve
the inequality.
6 times a number is greater than 4 times the number
minus 2.
Answer:
Solving Multi-Step Inequalities
When solving equations that contain grouping
symbols, first use the Distributive Property to
remove the grouping symbols.
Distributive Property
Original inequality
Distributive Property
Combine like terms.
Add c to each side.
Simplify.
Subtract 6 from each side.
Simplify.
Divide each side by 4.
Simplify.
Distributive Property
Answer: Since
is the same as
the solution set is
Do this one in your notes, please!
Distributive Property
3 p  2 p  8  p  2  3 p Distributive Property
p  8 4p  2
Combine like terms
8  3p  2
Subtract p from each side
10  3 p
Add 2 to each side
Answer:
Divide each side by 3
Solving Multi-Step Inequalities
If solving an inequality results in a statement
that is always true, the solution is all real
numbers.
If solving an inequality results in a statement
that is never true, the solution is the empty set
. The empty set has no members.
Empty Set
Original inequality
Distributive Property
Combine like terms.
Subtract 4s from each side.
This statement is false.
Answer: Since the inequality results in a false
statement, the solution set is the empty set Ø.
Do this one in your notes, please!
Empty Set
4r  6  2r  7r  8  5r Distributive Property
Combine like terms
2r  6  2r  8
 6  8
Answer: Ø
This statement is false.