RULES FOR OPERATIONS ON INEQUALITIES

Addition and subtraction properties of inequality :

When you add or subtract the same number from each side of an inequality, the inequality remains true.

These properties are also true for >, ≤ and ≥

To see more examples 

Multiplication and division properties of inequality :

When you multiply or divide each side of an inequality by the same positive number, the inequality remains true.

These properties are also true for >, ≤ and ≥

To see more examples

Multiplication and division properties of inequality :

Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign.

Solve the inequalities using addition and subtraction property :

Example 1 : 

y - 4 > -7

Solution : 

y - 4 > -7

Because 4 is subtracted from y, add 4 to each side to undo the subtraction.

(y - 4) + 4 > (-7) + 4

y - 4 + 4 > -3

y > -3

Example 2 : 

0.9 ≥ a - 0.2

Solution : 

0.9 ≥ a - 0.2

Because 0.2 is subtracted from a, add 0.2 to each side to undo the subtraction.

(0.9) + 0.2 ≥ (a - 0.2) + 0.2

0.9 + 0.2 ≥ a - 0.2 + 0.2

0.11 ≥ a

a ≤ 0.11

Example 3 : 

x + 7 < 13

Solution : 

x + 7 < 13

Because 7 is added to x, subtract 7 from each side to undo the addition.

(x + 7) - 7 < (13) - 7

x + 7 - 7 < 13 - 7

x < 6

Example 4 : 

b + 6.2 ≥ 9.2

Solution : 

b + 6.2 ≥ 9.2

Because 6.2 is added to b, subtract 6.2 from each side to undo the subtraction.

(b + 6.2) - 6.2 ≥ (9.2) - 6.2

b + 6.2 - 6.2 ≥ 9.2 - 6.2

b ≥ 3

Solve the inequalities using multiplication and division property :

Example 5 :  

y/2 > -1.5

Solution : 

y/2 > -1.5

Because y is divided by 2, multiply each side by 2 to undo division.

2(y/2) > 2(-1.5)

y > -3

Example 6 :  

-b/10 ≥ -0.11

Solution : 

-b/10 ≥ -0.11

Because -b is divided by 10, multiply each side by -10 and change ≥ to ≤. 

(-b/10)(-10) ≤ -0.11(-10)

10b/10 ≤ 1.1

b ≤ 1.1

Example 7 : 

3x < 18

Solution : 

3x < 18

Because x is multiplied by 3, divide each side by 3 to undo the multiplication.

3x/3 < 18/3

x < 6

Example 8 : 

-10a > 15

Solution : 

-10a > 15

Because a is multiplied by -10, divide each side by -10 and change > to <. 

-10a/(-10) < 15/(-10)

a < -1.5

Example 9 :

Jake plans to board his dog while he is away on vacation. Boarding house A charges $90 plus $5 per day. Boarding house B charges $100 plus $4 per day. For how many days must Jake board his dog for A to be less expensive than B?.

Solution : 

Let x be the number of days.

Charges of Boarding house A :

90 + 5x

Charges of Boarding house B :

100 + 4x

In house A, the expenses should be less.

90 + 5x < 100 + 4x

5x - 4x < 100 - 90

x < 10

Lesser than 10 days.

Example 10 :

The Student Council decides to raise money by organizing a dance. Tickets are $7.50 each, but he cost of hiring the video-DJ is $1200. How many tickets must be sold to make a profit of more than $1500?

Solution :

Let x be the number of tickets sold.

Cost of each ticket = $7.50

Profit should be more than 1500, then 

1200 + 7.50x > 1500

7.50x > 1500 - 1200

7.50 x > 300

x > 300/7.50

x > 40

So, more than 40 tickets.

Related pages

• Solve inequalities
  
• Writing and solving one step inequalities
  
• Writing and solving two step inequalities
• Graphing and writing inequalities

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