Properties of Inequalities
Addition property of Inequality
The addition property of inequality states that if a number is added to one side of an inequality, the same number must also be added to the opposite side so that the inequality remains satisfied.
Subtraction property of inequality
The subtraction property of inequality, just like the addition property, states that subtracting equal numbers from both sides of the inequality, does not change the inequality.
According to both these properties,
If a > b, then a + c > b + c, and a – c > b – c
If a < b, then a + c < b + c, and a – c < b – c
As an example of the addition property of inequality we see that even when 4 is added to both sides of the inequality, the inequality remains the same.
x – 4 >30
x – 4 + 4 > 30 + 4 (add 18 to both sides)
Hence, we see that the inequality remains the same, x > 34.
By graphing the inequality we get,
We can observe the subtraction property with this example. Notice that when 15 is subtracted from the inequality, the inequality remains unchanged.
x + 15 < 20
x + 15 – 15 < 20 – 15 (subtract 15 from both the sides)
Hence, the inequality remains the same, x < 5.
Graphing the inequality we get,
How do we model real life inequality problems?
Consider the following situation. To be a part of the junior basketball team, students should not be taller than 6.25 feet. Roger is 5 feet 6 inches tall. How much can he grow in height and still meet the requirement?
Roger’s current height = 5 feet 6 inches.
6 inches is 0.5 feet (1 foot is 12 inches)
Hence Roger’s height is 5.5 feet
Let the height Roger can grow be h
According to the question,
h + 5.5 6.25
h + 5.5 -5.5 6.25 -5.5 (Subtraction property of inequality)
h 0.75
This shows us that Roger can grow upto 0.75 feet and still be a part of the basketball team. The situation can be graphically represented as follows.
Learn More About Inequalities in This Video
Solved Examples on Addition and Subtraction of Inequalities
Question 1:
Solve the following inequality : x + 23 – 20 \(\geq\) 678
Solution:
x + 23 – 20 \(\geq\) 678
x + 3678
x + 3 – 3 \(\geq\) 678 -3 (Subtraction property of inequality)
x \(\geq\) 675
Question 2:
Seven more than a number is more than 50.
Solution:
Let the number be ‘x’
According to the question,
7 + x > 50
7 – 7 + x > 50 – 7 (Subtraction property of inequality)
x > 43
This means that the number under consideration is more than 43.
Question 3
The sum of a number and 39 is less than 50, what is the number?
Solution:
Let the number be x, the equation representing the question is as follows:
x + 39 < 50
x + 39 – 39 < 50 – 39 (Subtraction property of inequality)
x < 11
So, the number is less than 11.
Question 4
For a dance piece, a certain number of dancers were required. The requirement is that the final number must be less than 100. The choreographer has already chosen 12 dancers, how many dancers can be added to the troupe? Also, represent this situation graphically.
Let the number of dancers to be added be x.
Since 12 dancers were already chosen, the number of dancers is as follows
x + 12 < 100
Simplifying the equation, we get
x + 12 – 12 < 100 – 12 (Subtraction property of inequality)
x < 88
Therefore the number of dancers to be selected should be lesser than 88.
Question 5
Ice-cream was distributed for a family reunion. To understand the approximate quantity that was distributed, Alan remembers the following “From the total amount of ice cream, four scoops were distributed but the final quantity must be greater than or equal to two scoops.” Based on the statement, what was the minimum number of scoops that was initially present? Also plot a graph showing the same.
Solution:
Let the initial number of scoops of ice cream be x.
x – 4 \(\geq\) 2
Add 4 on both sides
x – 4 + 4 \(\geq\) 2 + 4 (Addition property of inequality)
x \(\geq\) 6
This means that a minimum of 6 scoops were initially available.
Now, graph the inequality
In equations, one side is equal to the other. In inequalities, one side of an inequality may or may not be equal to the other side.
Inequalities can be used to understand a minimum benchmark or a limit based on the circumstance. For example, if there is a height requirement for a ride at an amusement park, the situation can easily be expressed using an inequality.