Download Solve Multi-Step Inequalities

Solving and Graphing MultiStep Inequalities (5-3)
Objective: Solve linear inequalities
involving more than one operation.
Solve linear inequalities involving
the Distributive Property.
Solve Multi-Step Inequalities:




Multi-step inequalities can be solved by undoing the operations in
the same say you would solve a multi-step equation.
When multiplying or dividing by a negative number, the direction of
the inequality symbol changes. This holds true for multi-step
inequalities.
You can translate sentences into multi-step inequalities and then
solve them using the Properties of Inequalities.
When solving inequalities that contain grouping symbols, use the
Distributive Property to remove the grouping symbols first. Then
combine like terms, if possible, to simplify the resulting inequality.
Example 1

Adriana has a budget of $115 for faxes. The fax service she uses
charges $25 to activate an account and $0.08 per page to send
faxes. How many pages can Adriana fax and stay within her
budget?

She can spend at most $115 which means we must use ≤ in the inequality.
25 + 0.08p ≤ 115
-25
-25
0.08p ≤ 90
0.08 0.08
p ≤ 1125
She can send at
most 1125 pages.
Check Your Progress

Choose the best answer for the following.

Rob has a budget of $425 for senior pictures. The
cost for a basic package and sitting fee is $200.
He wants to buy extra wallet-size pictures for his
friends that cost $4.50 each. How many walletsize pictures can he order and stay within his
budget. Use the inequality 200 + 4.5p ≤ 425.
A.
B.
C.
D.
50 pictures
55 pictures
60 pictures
70 pictures
200 + 4.5p ≤ 425
-200
-200
4.5p ≤ 225
4.5
4.5
Example 2

Solve and graph the solution to 13 – 11d ≥ 79.
13 – 11d ≥ 79
-13
-13
-11d ≥ 66
-11 -11
d ≤ -6
{d|d ≤ -6}
Check Your Progress

Choose the best answer for
the following.
 Solve
-8y + 3 > -5.
{y|y < -1}
B. {y|y > 1}
C. {y|y > -1}
D. {y|y < 1}
A.
-8y + 3 > -5
-3 -3
-8y > -8
-8 -8
Example 3

Define a variable, write an inequality, solve the
problem, and graph the solution. Four times a number
plus twelve is less than the number minus three.

Let n = the number
4n + 12 < n - 3
-n
-n
3n + 12 < -3
-12 -12
3n < -15
3
3
n < -5
{n|n < -5}
Check Your Progress

Choose the best answer for the
following.

Write an inequality for the sentence
below. Then solve the inequality.
 6 times a number is greater than 4 times
the number minus 2.
6n > 4n – 2
A.
B.
C.
D.
6n > 4n – 2; {n|n > -1}
6n < rn – 2; {n|n < -1}
6n > 4n + 2; {n|n > 1}
6n > 2 – 4n; {n|n < -1/5}
-4n -4n
2n > -2
2
2
Example 4

Solve and graph the solution to 6c + 3(2 – c) ≥ -2c + 1.
6c + 3(2 – c) ≥ -2c + 1
6c + 6 – 3c ≥ -2c + 1
3c + 6 ≥ -2c + 1
+2c
+2c
5c + 6 ≥ 1
-6 -6
5c ≥ -5
5
5
c ≥ -1
{c|c ≥ -1}
Check Your Progress

Choose the best answer for the
following.

Solve 3p – 2(p – 4) < p – (2 – 3p).
A.
B.
C.
D.
{p|p > 10/3}
{p|p < 10/3}
{p|p > -10/3}
{p|p < -10/3}
3p – 2p + 8 < p – 2 + 3p
p + 8 < 4p – 2
-4p
-4p
-3p + 8 < -2
-8 -8
-3p < -10
-3 -3
Solve Multi-Step Inequalities

If solving an inequality results in a
statement that is always true, the solution
set is the set of all real numbers.
 This solution set is written as {x|x is a real
number}.
 If solving an inequality results in a
statement that is never true, the solution
set is the empty set, which is written as the
symbol .
 The empty set has no members.
Example 5

Solve each inequality.
a.
-7(k + 4) + 11k ≥ 8k – 2(2k + 1)
-7k – 28 + 11k ≥ 8k – 4k – 2
4k – 28 ≥ 4k – 2
-4k
-4k
-28 ≥ -2

(Not true!)
Example 5

Solve each inequality.
a.
2(4r + 3) ≤ 22 + 8(r – 2)
8r + 6 ≤ 22 + 8r – 16
8r + 6 ≤ 8r + 6
-8r
-8r
6≤6
(True!)
{r|r is a real number.}
Check Your Progress

Choose the best answer for the
following.
A.
Solve 8a + 5 ≤ 6a + 3(a + 4) – (a + 7).
8a + 5 ≤ 6a + 3a + 12 – a – 7
A. {a|a ≤ 3}
8a + 5 ≤ 8a + 5
B. {a|a ≤ 0}
-8a
-8a
C. {a|a is a real number.}
5≤5
(True!)
D. 
Check Your Progress

Choose the best answer for the
following.
B.
Solve 4r – 2(3 + r) < 7r – (8 + 5r).
4r – 6 – 2r < 7r – 8 – 5r
A. {r|r > 0}
2r – 6 < 2r – 8
B. {r|r < -1}
-2r
-2r
C. {r|r is a real number.}
-6 < -8
(Not True!)
D. 