Understanding Inequality Symbols | Grade 6 Math | LevelKite

You already know how to solve equations. An equation uses the equals sign to say that two quantities have the same value.

Many situations are not about being equal. A suitcase has to weigh less than a limit. A rider has to be at least a certain height. To write conditions like these, you need a new kind of statement.

An inequality is a mathematical statement that compares two quantities with a symbol such as $<$ , $>$ , $\le$ or $\ge$ . In this lesson you will learn what each of these four symbols means and how to read it.

Four inequality symbols

Symbol	How to read it	A true statement
$<$	is less than	$3 < 7$
$>$	is greater than	$7 > 3$
$\le$	is less than or equal to	$3 \le 7$ , and also $7 \le 7$
$\ge$	is greater than or equal to	$7 \ge 3$ , and also $7 \ge 7$

The shape of each symbol carries its meaning. Each one has a small pointed end and a wide open end.

With $<$ and $>$ , a statement is true when the pointed end points at the smaller quantity and the open end faces the larger one. In both $3 < 7$ and $7 > 3$ , the pointed end is next to $3$ and the open end faces $7$ . If the pointed end is next to the larger quantity, as in $7 < 3$ , the statement is false.
The line under $\le$ and $\ge$ is like a piece of an equals sign. It adds or equal to. Only one part has to hold: $3 \le 7$ is true because $3$ is less than $7$ , and $7 \le 7$ is true because the two sides are equal.

The symbols $<$ and $>$ have no line. A statement with one of them is false when the two quantities are equal.

Worked example

Decide whether a statement is true

Decide whether each statement is true or false: $9 > 4$ , $4 > 9$ , $6 \le 6$ and $6 < 6$ .

Read the first statement from left to right: 9 is greater than 4. That is correct, because 9 is 5 more than 4. The open end faces the larger number, so the statement is true.
$9 > 4 \quad \text{is true}$
Read the second statement the same way: 4 is greater than 9. That is not correct, because 4 is 5 less than 9. The open end faces the smaller number, so the statement is false.
$4 > 9 \quad \text{is false}$
Read the third statement: 6 is less than or equal to 6. The two sides are equal, and the line under the symbol allows equal, so the statement is true.
$6 \le 6 \quad \text{is true}$
Read the fourth statement: 6 is less than 6. This symbol has no line, so equal is not allowed. No number is less than itself, so the statement is false.
$6 < 6 \quad \text{is false}$

Answer

The statements $9 > 4$ and $6 \le 6$ are true. The statements $4 > 9$ and $6 < 6$ are false.

Now put a variable into an inequality. A variable is a letter that stands for a number. Let $x$ stand for a number. The inequality $x > 5$ says that $x$ is greater than $5$ .

Compare this with an equation. Only one value of $x$ makes $3x = 12$ true. You find it by dividing both sides by $3$ , which keeps the two sides equal: $12 \div 3 = 4$ , so $x = 4$ .

The inequality $x > 5$ does not pick out one value. It is true for $6$ , for $5.5$ and for $100$ , because each of these is greater than $5$ . The list of numbers greater than $5$ never ends, so the list of values that make $x > 5$ true never ends either.

The inequality is false when $x$ is $5$ , because $5$ is not greater than $5$ . It is also false when $x$ is less than $5$ .

Key concept

An inequality symbol compares left with right

Let $a$ and $b$ stand for two quantities. An inequality symbol tells you how $a$ , on the left, compares with $b$ , on the right. Read it from left to right, like a sentence.

$a < b$ says $a$ is less than $b$ .
$a > b$ says $a$ is greater than $b$ .
$a \le b$ says $a$ is less than $b$ or equal to $b$ .
$a \ge b$ says $a$ is greater than $b$ or equal to $b$ .

With $<$ and $>$ , the open end faces the quantity that the statement says is larger. With $\le$ and $\ge$ , the line means the two quantities are also allowed to be equal.

So $5 < x$ and $x > 5$ are two ways to write one comparison about a number $x$ . In both, the open end faces $x$ and the pointed end is next to $5$ .

That is why you may swap the two sides of an inequality, as long as you also turn the symbol around. The open end still faces the same quantity, so the meaning does not change. A line under the symbol stays: $x \le 9$ and $9 \ge x$ say the same thing.

If you swap the sides but leave the symbol alone, the meaning changes. The statement $5 > x$ says that $x$ is less than $5$ .

Worked example

Read an inequality with the variable on the right

Let $t$ stand for a number. What does $10 \ge t$ say about $t$ ? Which of the values $7$ , $10$ and $13$ make it true?

Read from left to right, starting with the left side: 10 is greater than or equal to t. The symbol compares the left side, 10, with the right side, t.
$10 \ge t$
Say the same thing starting with t. If 10 is greater than or equal to t, then t is less than or equal to 10. You swap the sides and turn the symbol around, so the open end still faces 10 and the meaning stays the same.
$t \le 10$
Test 7 by writing it in place of t. Since 10 is greater than 7, the statement is true.
$10 \ge 7 \quad \text{is true}$
Test 10 the same way. The two sides are equal, and the line under the symbol allows equal, so the statement is true.
$10 \ge 10 \quad \text{is true}$
Test 13. Here 10 is less than 13, so 10 is not greater than 13 and not equal to 13. The statement is false.
$10 \ge 13 \quad \text{is false}$

Answer

The inequality $10 \ge t$ says that $t$ is less than or equal to $10$ . The values $7$ and $10$ make it true, and $13$ does not.

Words tell you which symbol to use. The key question is whether the number you compare with is itself allowed.

At least $12$ means the least the number can be is $12$ . So $12$ is allowed, and so is every greater number. Use $\ge$ .
At most $12$ means the most the number can be is $12$ . So $12$ is allowed, and so is every smaller number. Use $\le$ .
More than $12$ and less than $12$ both leave $12$ out. Use $>$ for more than and $<$ for less than.
No less than $12$ rules out every number below $12$ , so it means the same as at least $12$ . No more than $12$ rules out every number above $12$ , so it means the same as at most $12$ .

In the table, $n$ stands for a number. The last column tests $n = 12$ .

Matching words to symbols

In words	Inequality	Is 12 itself allowed?
$n$ is more than $12$	$n > 12$	No. $12 > 12$ is false.
$n$ is less than $12$	$n < 12$	No. $12 < 12$ is false.
$n$ is at least $12$	$n \ge 12$	Yes. $12 \ge 12$ is true.
$n$ is no less than $12$	$n \ge 12$	Yes. $12 \ge 12$ is true.
$n$ is at most $12$	$n \le 12$	Yes. $12 \le 12$ is true.
$n$ is no more than $12$	$n \le 12$	Yes. $12 \le 12$ is true.

Worked example

Choose the symbol from the words

Let $h$ stand for a rider's height in inches. Sign A says, "You must be at least 48 inches tall." Sign B says, "You must be taller than 48 inches." Write an inequality for each sign. Then decide whether a rider who is exactly $48$ inches tall may ride under each sign.

Start with Sign A. At least 48 means the least height allowed is 48 inches, so a height equal to 48 is allowed and so is every greater height. Use the symbol with the line.
$h \ge 48$
Now Sign B. Taller than 48 means the height is greater than 48 inches. A height equal to 48 is not taller than 48, so use the symbol with no line.
$h > 48$
Test 48 under Sign A by writing it in place of h. The two sides are equal, and this symbol allows equal.
$48 \ge 48 \quad \text{is true}$
Test 48 under Sign B. The two sides are equal, but this symbol does not allow equal.
$48 > 48 \quad \text{is false}$

Answer

Sign A is $h \ge 48$ and Sign B is $h > 48$ , where $h$ is the rider's height in inches. A rider who is exactly $48$ inches tall may ride under Sign A but not under Sign B.

The takeaway: an inequality symbol compares the quantity on its left with the quantity on its right, so read it from left to right. In a true statement with $<$ or $>$ , the open end faces the larger quantity. The line under $\le$ and $\ge$ adds or equal to, so the two quantities may also be equal. Phrases such as at least and at most allow the number you compare with. Phrases such as more than and less than leave it out.

Next, you will graph inequalities on number lines and write inequalities from number lines.