1. Solving and Graphing Multi-Step Inequalities (5-3)
Objective: Solve linear inequalities involving more than one operation. Solve linear inequalities involving the Distributive Property.

2. 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.

3. Example 1 25 + 0.08p ≤ 115 She can send at most 1125 pages. -25 -25
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.
p ≤ 115
She can send at most 1125 pages.
0.08p ≤ 90
p ≤ 1125

4. 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 wallet-size pictures can he order and stay within his budget. Use the inequality p ≤ 425.
50 pictures
55 pictures
60 pictures
70 pictures
p ≤ 425
4.5p ≤ 225

5. Example 2 Solve and graph the solution to 13 – 11d ≥ 79. 13 – 11d ≥ 79
-11d ≥ 66
{d|d ≤ -6}
d ≤ -6

6. Check Your Progress Choose the best answer for the following.
Solve -8y + 3 > -5.
{y|y < -1}
{y|y > 1}
{y|y > -1}
{y|y < 1}
-8y + 3 > -5
-8y > -8

7. 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
{n|n < -5}
4n + 12 < n - 3
-n n
3n + 12 < -3
3n < -15
n < -5

8. 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; {n|n > -1}
6n < rn – 2; {n|n < -1}
6n > 4n + 2; {n|n > 1}
6n > 2 – 4n; {n|n < -1/5}
6n > 4n – 2
-4n -4n
2n > -2

9. Example 4 Solve and graph the solution to 6c + 3(2 – c) ≥ -2c + 1.
5c ≥ -5
c ≥ -1

10. Check Your Progress Choose the best answer for the following.
Solve 3p – 2(p – 4) < p – (2 – 3p).
{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 p
-3p + 8 < -2
-3p < -10

11. 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.

12. Example 5 Solve each inequality. -7(k + 4) + 11k ≥ 8k – 2(2k + 1) 
-28 ≥ -2
(Not true!) 

13. Example 5 Solve each inequality. 2(4r + 3) ≤ 22 + 8(r – 2)
6 ≤ 6
(True!)
{r|r is a real number.}

14. Check Your Progress Choose the best answer for the following.
Solve 8a + 5 ≤ 6a + 3(a + 4) – (a + 7).
{a|a ≤ 3}
{a|a ≤ 0}
{a|a is a real number.} 
8a + 5 ≤ 6a + 3a + 12 – a – 7
8a + 5 ≤ 8a + 5
-8a a
5 ≤ 5
(True!)

15. Check Your Progress Choose the best answer for the following.
Solve 4r – 2(3 + r) < 7r – (8 + 5r).
{r|r > 0}
{r|r < -1}
{r|r is a real number.} 
4r – 6 – 2r < 7r – 8 – 5r
2r – 6 < 2r – 8
-2r r
-6 < -8
(Not True!)