1. Solving Inequalities by Adding or Subtracting (SOL 7.15)
Course 2: Inequalities
Solving Inequalities by Adding or Subtracting (SOL 7.15)

2. Key Concept Addition Property of Inequalities
Words: If any number is added to each side of a true inequality, the resulting inequality is also true.
Symbols: For all numbers a, b, and c, the following are true;
If a > b, then a + c > b + c
If a < b, then a + c < b + c

3. Key Concept Subtraction Property of Inequalities
Words: If any number is subtracted from each side of a true inequality, the resulting inequality is also true.
Symbols: For all numbers a, b, and c, the following are true;
If a > b, then a - c > b - c
If a < b, then a - c < b - c

4. Addition and Subtraction Rules

5. Addition and Subtraction Properties
Examples:
2 < > 3
2 + 5 < – 2 > 3 – 2
7 < > 1

6. Solve an Inequality Using Subtraction
Solve y + 5 > 11
y + 5 – 5 > 11 – 5 (Subtract 5 from both sides)
y > 6 (Simplify)
Check: y + 5 > 11
7 + 5 > 11 (Replace y with 7 – a number > 6)
12 > 11 (This statement is true.)

7. Try it!
Solve a < 3
a + 9 < 3 (You can rewrite the inequality.)
a + 9 – 9 < 3 – 9
a < -6
Check: 9 + a < 3
< 3 (Replace “a” with -6 or less)
3 < 3
Why can you replace a with -6?

8. Solve an Inequality Using Addition
Solve x – 23 < 12
x – < (Add 23 to both sides)
x < 35 (This means all numbers less than or equal to 35)
Check: x – 23 < 12
< 12 (Replace x with 35)
12 < 12 (This statement is true.)

9. Solve an Inequality Using Addition
Solve > d – 8
> d – (Add 8 to each side)
-13 > d OR d < -13
Check > d – 8
-21 > -13 – 8
-21 > -21
Why can you use -13?

10. Try It! Solve a – 5 > 6 a – 5 + 5 > 6 + 5 a > 11
Can you use 11 to check your solution?
Check: a – 5 > 6
12 – 5 > 6
7 > 6

11. Graph Solutions of Inequalities
Solve h – 1.5 < 5
h – < (Add 1.5 to each side)
h < 6.5 (Simplify)
Graph the solution on a number line
If your variable is on the left, the inequality will point in the direction you should shade

12. Try It! Solve 33 < m – (-6) 33 < m + 6 (Simplify)
m + 6 > 33 (You can rewrite it with the variable on the left.)
m + 6 – 6 > 33 – 6 (Subtract 6 from each side)
m > 27
Graph the solution on a number line.
Place a closed circle on the number line on the number 27
Shade to the right (positive) side

13. Graph Solutions of Inequalities
Solve 33 < m – (-6)
33 < m + 6 (Simplify)
m + 6 > 33 (You can rewrite it with the variable on the left.)
m + 6 – 6 > 33 – 6 (Subtract 6 from each side)
m > 27
Graph the solution on a number line

14. Use an Inequality to Solve a Problem
Katya has $12 to take to the bowling alley. If the shoe rental costs $3.75, what is the most she can spend on games and snacks?
“The most” means “no more than” or “less than or equal to”
Cost of shoe rental + games and snacks must be less than or equal to $12.
$ c < $12
$ c - $3.75 < $12 - $3.75
c < $8.25
Katya can spend no more than $8.25.

15. Try It!
Chris is saving money for a ski trip. He has $62.50, but his goal is to save at least $100. What is the least amount Chris needs to save to reach his goal?
Current amount + money saved must be greater than or equal to $100
$ s > $100
$ s - $62.50 > $100 - $62.50
s > $37.50
Chris must save at least $37.50.