Recognizing Inequalities - First Grade Mathematics

What happens when two amounts do NOT balance? In the real world, two groups are often unequal: one person has more crayons, one basket holds fewer apples, or one tower of blocks is taller than another. In mathematics, when two quantities are not equal, we call that relationship an inequality! The prefix "in-" means "not," so an inequality simply means "not equal." In this chapter, you will learn the not-equal symbol, master the greater-than and less-than inequality symbols, and learn how to recognize and write true inequality statements.

The Not-Equal Symbol: ≠

The simplest way to state that two amounts are not balanced is to draw an equal sign and strike a diagonal slash right through it!

The Not-Equal Symbol (≠):
        ===\===
        ===/===
Means: "IS NOT EQUAL TO", "DOES NOT BALANCE"

Example:  7 ≠ 4   ("Seven is not equal to four!")

The Slash Means "No!"

Just like a red circle with a slash through it means "no parking" or "do not enter," an equal sign with a slash through it means: "These two sides do NOT have the same value!" 5 + 2 ≠ 8 (because 7 does not equal 8).

The Greater-Than and Less-Than Inequality Symbols

While ≠ tells you that two sides are unequal, mathematicians prefer to use symbols that tell you WHICH side is bigger!

The Two Inequality Direction Symbols:

   >   GREATER THAN
       8 > 3   ("Eight is greater than three")
       Open mouth faces the larger number 8!

   <   LESS THAN
       2 < 9   ("Two is less than nine")
       Pointed tip pokes the smaller number 2!

The Hungry Creature Trick

Remember: the open mouth ALWAYS opens wide toward the bigger quantity! If the big number is on the left, use > (e.g., 10 > 6). If the big number is on the right, use < (e.g., 4 < 7).

Evaluating Inequalities with Operations

Inequalities can also compare math expressions with addition or subtraction!

Testing:  5 + 3  [ ? ]  10

Step 1: Calculate the left expression:
        5 + 3 = 8

Step 2: Compare 8 to 10:
        8 is LESS THAN 10!
        Use the < symbol!

Statement:  5 + 3 < 10  ("Five plus three is less than ten!")

True vs. False Inequalities

Just like equations, inequalities can be true or false: 6 + 2 > 5 is TRUE (8 is greater than 5). 4 + 1 > 9 is FALSE (5 is NOT greater than 9; 5 is less than 9!).

Practice Exercises

Exercise 1: What does the prefix "in-" mean in the word "inequality"? What does the symbol ≠ mean?

Exercise 2: Write the correct symbol (<, >, or =) in each circle: 7 [ ] 12 15 [ ] 9 4 + 4 [ ] 8 6 + 3 [ ] 11

Exercise 3: Is the inequality statement 8 - 2 > 4 true or false? Explain why by evaluating the left side.

Exercise 4: Write an inequality statement comparing the number of sides on a triangle to the number of sides on a hexagon using the < symbol.

Solutions and Explanations

Solution to Exercise 1: The prefix "in-" means "not," so "inequality" means not equal. The symbol ≠ means "is not equal to."

Solution to Exercise 2: 7 < 12 15 > 9 4 + 4 = 8 6 + 3 < 11 (9 < 11)

Solution to Exercise 3: The statement 8 - 2 > 4 is TRUE. Left side: 8 - 2 = 6. Since 6 is greater than 4, 6 > 4 is completely true.

Solution to Exercise 4: A triangle has 3 sides and a hexagon has 6 sides: 3 < 6.