About Properties of Inequality
What are Inequalities?
An inequality is a mathematical correlation between the two expressions that can be expressed using any of the following symbols:
- \( \leq \) : ‘less than or equal to’
- < : ‘less than’
- \( \neq \) : ‘not equal to’
- > : ‘greater than’:
- \( \geq \) : ‘greater than or equal to’
In a nutshell, an inequality compares two values that are unequal to the other.
For example, writing a sentence as an inequality.
1. The value of the \(y\) is less than 6
\( \Rightarrow y< 6 \)
2. The value of \(x + 3\) is greater than or equal to 6
\(\Rightarrow (x+3)\geq 6 \)
What are Properties of Inequalities?
Here are the list of Properties of Inequalities:
1. Additive property of inequality:
Adding the same value to both sides of the inequality does not change the inequality.
- If p < q, then p + r < q + r
- If p > q, then p + r > q + r
- If p \( \leq \) q, then p + r \( \leq \) q + r
- If p \( \geq \) q, then p + r \( \geq \) q + r
For example,
Numerical form:
5 < 9
5 + 2 < 9 + 2 (Adding 2 on both sides)
7 < 11
Algebraic form:
\(y-3 > 2\)
\(y-3 + 3 > 2 + 3\) (Adding 3 on both sides)
\(y>5\)
2. Subtraction property of inequality:
Subtracting the same value from both sides of the inequality does not change the inequality.
- If p < q, then p – r < q – r
- If p > q, then p – r > q – r
- If p \( \leq \) q, then p – r \( \leq \) q – r
- If p \( \geq \) q, then p – r \( \geq \) q – r
For example,
Numerical form:
3 < 6
3 – 2 < 6 – 2 (subtracting 2 on both sides)
1 < 4
Algebraic form:
\(x+5>11\)
\(x+5-5>11-5\) (subtracting 5 on both sides)
\(x>6\)
3. Multiplication property of inequality:
Let’s take two positive numbers p and q. The inequality remains the same when both p and q are multiplied by the same positive number. When both p and q are multiplied by the same negative number, however, the inequality reverses.
- If p < q and if r is a positive number, then p \( \times \) r < q \( \times \) r.
- If p < q and if r is a negative number, then p\( \times \) r > q \( \times \) r
For example,
Numerical form:
7 < 9
7\(\times\)2 < 9\(\times\) 2 (multiplying 2 on both sides)
14 < 18
Algebraic form:
\( \frac{x}{5}> 4 \)
\( \frac{x}{5}\times 5 > 4\times 5 \) (multiplying 5 on both sides)
\(x>20\)
4. Division property of inequality:
Let’s take two positive numbers p and q. The inequality remains the same when both p and q are divided by the same positive number. When both p and q are divided by the same negative number, however, the inequality reverses.
- If p < q and if r is a positive number, then \( \frac{p}{r}< \frac{q}{r} \)
- If p < q and if r is a negative number, then \( \frac{p}{r}> \frac{q}{r} \)
For example,
Numerical form:
6 < 8
6 \( \div \) 2 < 8 \( \div \) 2 (dividing 2 on both sides)
3 < 4
Algebraic form:
\(2x>4\)
\( \frac{2x}{2}> \frac{4}{2} \) (dividing by 2 on both sides)
\(x>2\)
Learn More About Inequalities in This Video
Solved Examples on Properties of Inequalities
Example 1: A taxi charges a flat rate of $0.75 and furthermore $0.35 per mile. John can only afford to spend $6 on a ride. Form an inequality that represents John’s scenario. How far can John travel without going over his budget? Justify your response.
Solution:
Let M denote the number of miles he traveled.
Writing the inequality, 0.35M + 0.75 \( \leq \) 6
Where,
- 0.35M is 0.35 per mile and here, per mile means multiply.
- ‘+’ means addition.
- 0.75 is flat rate.
- ‘\( \leq \)’ means no more than.
- ‘6’ means money to spend.
No more than’ means you can’t have more than one of something, so you must have fewer.
Solving the inequality:
0.35M + 0.75 \( \leq \) 6
0.35M + 0.75 – 0.75 \( \leq \) 6 – 0.75 (subtracting 0.75 on both sides)
0.35M \( \leq \) 5.25 (simplified)
\( \frac{0.35M}{0.35}\leq \frac{5.25}{0.35} \) (Divided both sides with 0.35)
M \( \leq \) 15 (simplified)
Representing on number line:
John can travel less than or equal to 15 miles before he reaches his limit of $6.
Check: 0.35M + 0.75 \( \leq \) 6
0.35 \( \times \) (15) + 0.75 \( \leq \) 6 (substituted 15 in the place of M)
5.25 + 0.75 \( \leq \) 6 (Multiplied)
6 \( \leq \) 6 (simplified)
Hence the answer is justified.
Example 2: For a birthday party rental, Sue’s Party Planning Co. charges a flat fee of $30 plus $3.50 per person. Joseph has only $90 to spend on his birthday celebration. To represent Joseph’s situation, create an inequality. Without going over his limit, how many people can Joseph invite to his birthday party?
Solution:
Let P represent the number of people.
Writing the inequality, 3.50p + 30 \( \leq \) 90
Where,
- 3.50p is 3.50 dollars per person.
- 30 is flat rate.
- ‘ \( \leq 90 \)’ implies no more than 90 to spend
Solving the inequality:
3.50p+30 \( \leq \) 90
3.50p + 30 – 30 \( \leq \) 90 – 30 (subtracting 30 on both sides)
3.50p \( \leq \) 60 (simplified)
\( \frac{3.50p}{3.50}\leq \frac{60}{3.50} \) (Divided both sides with 3.50)
p \( \leq \) 17.14
Representing on number line:
Hence Joseph can invite a maximum of 17 people to his birthday party.
Example 3: A liter of orange juice will cost you at least $1.50. What is the amount of juice you can purchase for $12?
Solution:
Let, \(x=\) quantity of juice
Writing the inequality, \(1.50x \leq \) 12
Solving the inequality:
\( 1.50x \leq \) 12
\( \frac{1.50x}{1.50}\leq \frac{12}{1.50} \) (Divided both sides with 1.50)
\(x \leq \) 8
Representing on a number line:
The situation can be represented by \(x \leq \) 8. This implies that a maximum of 8 liters of juice can be purchased.
Inequalities can be used in our day to day life in situations where we may not know the exact value but may know the approximate value. For example, I have less than $20 with me, there are more than 100 people in the stadium, etc.
Let’s understand this with the help of an example.
3 < 10
Multiplying -2 on both sides
\(3\times -2 < 10\times -2 \)
\(-6<-20\)
Now we know that this inequality is incorrect
So, \(-6>-20\).
Hence on multiplying both sides by the same negative number the inequality reverses.