1. Section 6.3 “Solve Multi-Step Inequalities”
mathematical sentence formed by placing a <, ≤, >, or ≥ between two expressions.
13 * 22 ≥ 14 - z
2y – 8 < 12
5c + 14 > 30
w/ ≤ 121

2. Section 6.3 “Solve Multi-Step Inequalities”
The steps for solving two-step and multi-step
equations can be applied to linear inequalities.
Solving Multi-Step Inequalities
STEP 1- Use distributive property and combine like terms.
STEP 2- Collect variables on one side of the inequality.
STEP 3- “Undo” addition and/or subtraction.
STEP 4- “Undo” multiplication and/or division.
STEP 5- Solve for the variable.
STEP 6- Check your work.

3. REMEMBER!!!!!
Multiplying and/or dividing each side of an inequality by a NEGATIVE number only produces an equivalent inequality IF the inequality sign is REVERSED!!

4. 3x – 7 < 8. Graph your solution. Solve
Write original inequality.
3x < 15
Add 7 to each side.
x < 5
Divide each side by 3.
ANSWER
The solutions are all real numbers less than 5. Check by substituting a number less than 5 in the original inequality.

5. Solve – 0.6(x – 5) < 15 – –0.6(x – 5) < 15 – –0.6x + 3 < 15 –
Write original inequality.
–0.6x < 15 –
Distributive property
– 0.6x < 12 –
Subtract 3 from each side. –
x >
–20
Divide each side by Reverse
inequality symbol. –
ANSWER
The solutions are all real numbers greater than equal Check by substituting a number more than -20 in the original inequality.

6. –1 ( p –12) > –2 4 –1 ( p –12) > -2 4 – ¼ p + 3 > -2
Write original inequality.
– ¼ p + 3 > -2
Distributive property.
-¼ p > –5
p < 20
ANSWER
The solutions are all real numbers greater than 20. Check by substituting a number greater than 20.

7. 6x – 7 > 2x+17. Graph your solution. Solve
Write original inequality.
6x > 2x+24
Add 7 to each side.
4x > 24
Subtract 2x from each side.
x > 6
Divide each side by 4.
ANSWER
The solutions are all real numbers greater than 6.

8. There are no solutions because 5 < – 21 is false. 5 < – 21
Solve: 14x + 5 < 7(2x – 3)
14x + 5 < 7(2x – 3)
Write original inequality.
14x + 5 < 14x – 21
Distributive property
There are no solutions because 5 < – 21 is false.
5 < – 21
Subtract 14x from each side.
**HINT**
If an inequality is equivalent to an inequality that is false,
such as 5 < -21, then the solution of the inequality has
NO SOLUTION.

9. All real numbers are solutions because – 1 > – 6 is true.
12x – 1 > 6(2x – 1)
12x – 1 > 6(2x – 1)
Write original inequality.
12x – 1 > 12x – 6
Distributive property
All real numbers are solutions because – 1 > – 6 is true.
– 1 > – 6
Subtract 12x from each side.
**HINT**
If an inequality is equivalent to an inequality that is true,
such as -1 > -6, then the solutions of the inequality are
ALL REAL NUMBERS .

10. Graphs of “No Solution” and “All Real Numbers”

11. Car Wash STEP 1 + < • 1.99 + g 8 20 < –
Use the sign shown. A gas station charges $.10 less per gallon of gasoline if a customer also gets a car wash. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most $20?
Because you are getting a car wash, you will pay $2.09 –$.10 = $1.99 per gallon of gasoline. Let g be the amount (in gallons) of gasoline that you buy.
STEP 1
Write a verbal model. Then write an inequality.
Price of gasoline (dollars/gallon)
Amount of gasoline (gallons)
Price of car wash (dollars)
Maximum amount (dollars) +
< •
1.99 + g 8 20
< –

12. STEP 2
Solve the inequality.
1.99g + 8 ≤ 20
Write inequality.
Subtract 8 from each side.
1.99g ≤ 12
Divide each side by 1.99.
You can buy up to slightly more than 6 gallons of gasoline.
g ≤
CHECK
Gasoline
(gal)
Total amount spent
(dollars)
8.00 1
9.99 2
11.98 3
13.97 4
15.96 5
17.95 6
19.94
You can use a table to check the reasonableness of your answer.The table shows that you will pay $19.94 for exactly 6 gallons of gasoline. Because $19.94 is less than $20, it is reasonable to conclude that you can buy slightly more than 6 gallons of gasoline.