Topics
6. Equations & Inequalities
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United States of AmericaVirginia
Grade 6

6.05 Write inequality statements

Lesson
Practice

Write inequality statements

An inequality is a mathematical statement that compares the size of two values. Inequalities use symbols such as:

Greater than \left(\gt\right)

means that the value to the left of the symbol is larger than the value to the right of the symbol

Example:

3\gt 2 means "3 is greater than 2"

Greater than or equal to \left(\geq\right)

means that the value to the left of the symbol is larger than or the same as the value to the right of the symbol

Example:

3\geq2, as well as 2\geq2, or n\geq2

Less than \left(\lt\right)

means that the value to the left of the symbol is smaller than the value to the right of the symbol

Example:

3\lt 4 means "3 is less than 4"

Less than or equal to \left(\leq\right)

means that the value to the left of the symbol is smaller than or the same as the value to the right of the symbol.

Example:

3\leq4, as well as 4\leq 4, or n\leq4

The images below show another demonstration of the inequality symbols:

An image showing less than, equal, and greater than symbol with small squares on it. Ask your teacher for more information

We are familiar with being able to write an equation in two orders. For example, x=10 and 10=x mean the same thing.

Every inequality can be written in two ways, but we need to be careful about the symbols we use.

In the following image, there are five cats on the left and three cats on the right. We would say that five cats is greater than three cats. We can write this as an inequality 5\gt 3.

A picture showing 5 cats on the left, a 'greater than' symbol at the middle and 3 cats on the right.

If we switch the order so that there are three cats on the left and five cats on the right, we can say that three cats is less than five cats. We can write this as an inequality 3\lt 5.

A picture showing 3 cats on the left, a 'less than' symbol at the middle and 5 cats on the right.

Both images mean the same thing but are stated differently. If we switch the order of an inequality, we have to change the inequality sign. This is also true with algebraic inequalities.

For example, x\gt 10 means the same thing as 10\lt x. In other words, "x is greater than ten" is the same as "ten is less than x".

For example, the expressions x \gt 5 and 5 \gt x represent different sets of numbers, while x \gt 5 and 5 \lt x represent the same set of numbers.

We can use this understanding of inequality symbols to write inequalities that represent real world situations. Let's write an inequality to represent the statement: "a student needs to score at least 75 points to pass an exam."

Let s represent the student's score. The key phrase "at least 75 points" tells us that the lowest passing score is 75. So the student will pass the exam if they score 75 points or if they score more than 75 points. If we use s to represent the score we can write the inequality s \geq 75.

Here are some common phrases and examples used for the different inequality symbols.

Inequality SymbolVocabulary/RepresentationsExample
\enspace\enspace\enspace\enspace\enspace\enspace \lt\text{less than, fewer than, under}\text{"The speed limit is less than 60 mph."} \\\ \text{translates to } S \lt 60
\enspace\enspace\enspace\enspace\enspace\enspace\gt\text{greater than, exceeds, more than}\text{"The temperature is greater than } 30\degree \text{C"} \\\ \text{translates to } T \gt 30
\enspace\enspace\enspace\enspace\enspace\enspace\leq\text{less than or equal to, at most,}\\\ \text{ no more than, up to}\text{"You can spend up to 50 dollars."}\\\ \text{translates to } C \leq 50
\enspace\enspace\enspace\enspace\enspace\enspace\geq\text{greater than or equal to, at least,} \\\ \text{ no less than}\text{"You need at least 8 hours of sleep."}\\\ \text{ translates to }H \geq 8

Examples

Example 1

For the number sentence \dfrac{2}{3} \enspace ⬚ \enspace 0.3

a

Choose the mathematical symbol that makes the number sentence true.

Worked Solution
Create a strategy

Convert the fraction to a decimal to compare values easily.

Apply the idea
\displaystyle \frac{2}{3}\displaystyle =\displaystyle 0.\overline{6}Convert to decimal
\displaystyle 0.\overline{6}\displaystyle >\displaystyle 0.3Compare the decimals

So we can see that \dfrac{2}{3}\gt 0.3.

Reflect and check

Another true inequality with the same numbers is 0.3 \lt \dfrac23.

b

Write the statement in words.

Worked Solution
Create a strategy

Replace the inequality symbol \gt with a vocabulary that has the same meaning.

Apply the idea

"\dfrac{2}{3} is greater than 0.3" or "\dfrac{2}{3} is more than 0.3"

Example 2

Write an inequality to represent each of the following situations.

a

n is greater than 9

Worked Solution
Create a strategy

The phrase "greater than" tells us which inequality symbol to use.

Apply the idea

n\gt 9

b

The weight of the package is under 5\text{ kg}. Let w be the weight of the package.

Worked Solution
Create a strategy

The word "under" means less than.

Apply the idea

w \lt 5

c

You must be at least 18 years old to vote in the United States. Let a be the age of the voters.

Worked Solution
Create a strategy

The phrase "at least 18" means you can vote in the US if you are exactly 18 years old or if you are older.

Apply the idea

a \geq 18

d

The maximum height for the ride is 120\text{ cm}. Let h be the height of the riders.

Worked Solution
Create a strategy

The word "maximum" means highest. So to ride you can be that height or shorter but you cannot be taller.

Apply the idea

h \leq 120

Example 3

Write a real-world scenario for each inequality.

a

x \leq 20

Worked Solution
Create a strategy

We can think of x as an object that can contain up to 20 pieces of another object.

Apply the idea

A sample scenario would be "A group of students is preparing boxes for toy donation. Each box can hold up to 20 small toys".

b

y \geq 5

Worked Solution
Create a strategy

We can write a scenario where y represents an activity that requires 5 or more of something.

Apply the idea

A sample scenario would be "A boat ride requires at least 5 passengers before it sails".

c

a \lt 0

Worked Solution
Create a strategy

We can think of a scenario where a represents an event that requires a number smaller than 0.

Apply the idea

A sample scenario would be "For water to start freezing, the temperature should be less than 0\degree \text{C}".

d

s \gt 20

Worked Solution
Create a strategy

We can write a scenario where s represents an event or result that must be larger than 20.

Apply the idea

A sample scenario would be "On a quiz, getting a score of greater than 20 would result in a pass".

Example 4

A clothing store has a fitting room policy that limits the number of garments one can bring in at a time. The policy can be represented by s \leq 4, where s represents the number of garments. Which options show the number of garments you can try at once? Select all correct options.

A
1
B
2
C
3
D
4
E
5
F
6
Worked Solution
Create a strategy

The symbol \leq means that the expression on the left of the symbol has a smaller value or the same value as the number on the right of the symbol. So, you can try on 4 garments or less.

Apply the idea

We can compare each number of garments to see which are less than or equal to 4. If the inequality is true, the option is correct, if the statement is fales, the option is incorrect.

Since 1 \leq 4 is a true statement, A is a correct option.

Since 2 \leq 4 is a true statement, B is a correct option.

Since 3 \leq 4 is a true statement, C is a correct option.

Since 4 \leq 4 is a true statement, D is a correct option.

Since 5 \leq 4 is a false statement, E is a incorrect option.

Since 6 \leq 4 is a false statement, F is a incorrect option.

So, the correct options are A, B, C and D.

Reflect and check

Let's check s \leq 4 on a number line.

The number line shows that all values to the left of 4, and including 4 are all possible values of s.

123456

However, in the given context, s represents the number of garments, which means that s should have a minimum value of 0, because it is not possible to try on a negative number of garments.

Idea summary

Inequalities are mathematical sentences where two expressions are not necessarily equal, indicated by the symbols: \lt , \gt , \leq, and \geq.

SymbolMeaningExample
\enspace \enspace\, \lt\text{less than, fewer than, under}\enspace 3\lt 6
\enspace \enspace\, \gt\text{greater than, exceeds, more than}\enspace 6\gt 3
\enspace \enspace\, \leq\text{less than or equal to, at most, no more than, up to}\enspace 4\leq 6
\enspace \enspace\, \geq\text{greater than or equal to, at least, no less than}\enspace 6\geq 5

Outcomes

6.PFA.4

The student will represent a contextual situation using a linear inequality in one variable with symbols and graphs on a number line.

6.PFA.4b

Write a linear inequality in one variable to represent a given constraint or condition in context or given a graph on a number line.

6.PFA.4c

Given a linear inequality in one variable, create a corresponding contextual situation or create a number line graph.

6.PFA.4e

Identify a numerical value(s) that is part of the solution set of a given inequality in one variable.

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