Writing Inequalities for Real-World Constraints | Grade 6 Math | LevelKite

A sign at a ride says how tall you must be. A carry-on bag has a weight limit. A ticket costs less for children under a certain age. Each of these is a constraint: a rule that limits the values a quantity is allowed to have.

To write a constraint in symbols, you use a variable: a letter that stands for a quantity that can change. Suppose $h$ stands for a rider's height in inches. The statement $h = 48$ allows only one height, exactly $48$ inches. A height rule for a ride allows a whole range of heights. To describe a range, you write an inequality: a statement that uses a symbol such as $>$ , $<$ , $\ge$ or $\le$ to compare two quantities.

In this lesson, you will turn constraints written in words into inequalities.

You will use four inequality symbols. The table below shows exactly what each one says. In the table, $x$ is a variable and $5$ is the number it is compared with.

Look for the extra line under $\ge$ and $\le$ . Think of it as part of an equals sign. It tells you that a value equal to the number is allowed too.

The four inequality symbols

Inequality	How to read it	Values that make it true	Is $5$ included?
$x > 5$	$x$ is greater than $5$	Numbers greater than $5$ , such as $5.1$ , $6$ and $40$	No
$x < 5$	$x$ is less than $5$	Numbers less than $5$ , such as $4.9$ , $4$ and $0$	No
$x \ge 5$	$x$ is greater than or equal to $5$	$5$ itself and numbers greater than $5$	Yes
$x \le 5$	$x$ is less than or equal to $5$	$5$ itself and numbers less than $5$	Yes

Every constraint in this lesson has a boundary number: the number where the allowed values start or stop. To write the inequality, you look at the values on each side of that number.

Try this rule: "A carry-on bag must weigh less than $20$ pounds." Let $w$ be the weight of the bag in pounds. The boundary number is $20$ .

A bag of $12$ pounds is allowed. So is a bag of $19.5$ pounds.
A bag of $23$ pounds is not allowed.
A bag of exactly $20$ pounds is not allowed, because $20$ is not less than $20$ .

The allowed weights are on the less side of $20$ , and $20$ itself is left out. The symbol that says this is $<$ , so the inequality is $w < 20$ .

Key concept

Two questions choose the symbol

To write an inequality for a constraint, name the quantity with a variable and state its unit. Find the boundary number. Then ask two questions.

Which side is allowed? Are the allowed values greater than the boundary number or less than it?
Is the boundary number itself allowed?

Your two answers choose the symbol:

Greater side, boundary not allowed: $>$
Less side, boundary not allowed: $<$
Greater side, boundary allowed: $\ge$
Less side, boundary allowed: $\le$

Write the variable, then the symbol, then the boundary number. This method fits a constraint that has one boundary number and allows values on just one side of it. Every constraint in this lesson is like that.

Constraints often use the same few phrases. The table below shows what these phrases mean in math problems. In the table, $x$ stands for the quantity and $12$ is the boundary number.

Read the whole phrase, not just one word. The word more is in both "more than $12$ " and "no more than $12$ ", but the two phrases need different symbols. If a constraint uses words that are not in the table, go back to the two questions.

Common phrases and the inequalities they describe

Phrase in the constraint	Which side is allowed?	Is $12$ allowed?	Inequality
more than $12$ , greater than $12$ , over $12$ , above $12$	Greater	No	$x > 12$
less than $12$ , fewer than $12$ , under $12$ , below $12$	Less	No	$x < 12$
at least $12$ , no less than $12$ , a minimum of $12$	Greater	Yes	$x \ge 12$
at most $12$ , no more than $12$ , a maximum of $12$	Less	Yes	$x \le 12$

Worked example

Example 1: more than

To win a prize in the school read-a-thon, a student must read for more than $300$ minutes. Write an inequality for the reading times that win a prize.

Name the quantity and its unit first, so the inequality has a clear meaning. Let m be the number of minutes the student reads.
Find the boundary number. The rule compares the reading time with 300 minutes, so the boundary number is 300.
Ask which side is allowed. A time of 450 minutes is more than 300, so it wins a prize. A time of 250 minutes does not. The allowed times are on the greater side of 300.
Ask whether 300 itself is allowed. Exactly 300 minutes is not more than 300, so it does not win. The greater side with the boundary not allowed means "greater than". Write the variable, then the symbol, then the boundary number.
$m > 300$
Check with three test values: 450 should win, and 250 and 300 should not. Put each one in place of m. The inequality agrees with the rule every time.
$450 > 300 \text{ is true}, \quad 250 > 300 \text{ is false}, \quad 300 > 300 \text{ is false}$

Answer

The inequality is $m > 300$ , where $m$ is the number of minutes the student reads. A student wins a prize for any reading time greater than $300$ minutes.

Worked example

Example 2: no more than

Each talk at the science fair may last no more than $8$ minutes. Write an inequality for the talk lengths that follow this rule.

Name the quantity and its unit. Let t be the length of a talk in minutes.
Find the boundary number. The rule compares the length of a talk with 8 minutes, so the boundary number is 8.
Ask which side is allowed. Read the whole phrase: "no more than 8" rules out every length greater than 8. A talk of 10 minutes breaks the rule and a talk of 6 minutes follows it. The allowed lengths are on the less side of 8.
Ask whether 8 itself is allowed. A talk of exactly 8 minutes is not more than 8 minutes, so it follows the rule. The less side with the boundary allowed means "less than or equal to".
$t \le 8$
Check with three test values: 6 and 8 should follow the rule, and 10 should not. The inequality agrees with the rule every time.
$6 \le 8 \text{ is true}, \quad 8 \le 8 \text{ is true}, \quad 10 \le 8 \text{ is false}$

Answer

The inequality is $t \le 8$ , where $t$ is the length of a talk in minutes. A talk follows the rule when it lasts $8$ minutes or less.

Worked example

Example 3: words that are not in the table

A board game box says: "Ages $8$ and up. Playing time: $30$ minutes." Write an inequality for the ages of the players the game is made for.

Name the quantity and its unit. The question asks about ages, so let y be a player's age in years.
Find the boundary number. The 30 minutes tells you how long a game lasts. It does not limit a player's age. The number that limits the age is 8.
Answer the two questions. "And up" puts the allowed ages on the greater side of 8. The box names 8 itself, so 8 is allowed. The greater side with the boundary allowed means "greater than or equal to".
$y \ge 8$
Check with three test values: 11 and 8 should be allowed, and 5 should not. The inequality agrees with the box every time.
$11 \ge 8 \text{ is true}, \quad 8 \ge 8 \text{ is true}, \quad 5 \ge 8 \text{ is false}$

Answer

The inequality is $y \ge 8$ , where $y$ is a player's age in years. The game is made for players who are $8$ years old or older.

A constraint is a rule about which values a quantity may have. To write it as an inequality, name the quantity with a variable and its unit, and find the boundary number. Then ask two questions: which side of the boundary number is allowed, and is the boundary number itself allowed? Your answers choose one of $>$ , $<$ , $\ge$ or $\le$ . Read whole phrases, not single words, and check your inequality with three test values.

Next, you will write and graph inequalities for word problems, see how many values can make an inequality true, and learn to spot mistakes in writing and graphing inequalities.