Course 2: Inequalities

Solving Inequalities by Adding or Subtracting (SOL 7.15)

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

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

Addition and Subtraction Rules

Addition and Subtraction Properties

Examples: 2 < 4 6 > 3

2 + 5 < 4 + 5 6 – 2 > 3 – 2

7 < 9 4 > 1

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.)

Try it!

Solve 9 + a < 3

a + 9 < 3 (You can rewrite the inequality.)

a + 9 – 9 < 3 – 9

a < -6 Check: 9 + a < 3

9 + -6 < 3 (Replace “a” with -6 or less)

3 < 3

Why can you replace a with -6?

Solve an Inequality Using Addition

Solve x – 23 < 12

x – 23 + 23 < 12 + 23 (Add 23 to both sides)

x < 35 (This means all numbers less than or equal to 35)

Check: x – 23 < 12

35 -23 < 12 (Replace x with 35)

12 < 12 (This statement is true.)

Solve an Inequality Using Addition

Solve -21 > d – 8

-21 + 8 > d – 8 + 8 (Add 8 to each side)

-13 > d OR d < -13 Check -21 > d – 8

-21 > -13 – 8

-21 > -21

Why can you use -13?

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

Graph Solutions of Inequalities

Solve h – 1.5 < 5

h – 1.5 + 1.5 < 5 + 1.5 (Add 1.5 to each side)

h < 6.5 (Simplify)

Graph the solution on a number line

5 6 7 8

If your variable is on the left, the inequality will point in the direction you should shade

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

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

26 27 28 29

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.

$3.75 + c < $12 $3.75 + c - $3.75 < $12 - $3.75 c < $8.25Katya can spend no more than $8.25.

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

$62.50 + s > $100 $62.50 + s - $62.50 > $100 - $62.50 s > $37.50Chris must save at least $37.50.