Expression, equation and inequality
The cost of four cones is a number. “Four cones cost 12 dollars” is a claim that could be wrong. How do you tell the two apart once they are written in symbols?
Where you are 5 in Variables
Why this exists
At the ice cream van a cone costs 3 dollars. “The cost of four cones” names an amount: 4 \cdot 3 dollars. “Four cones cost 12 dollars” does something else. It says something about that amount, and you can check whether it is right. Change the 12 to 13 and you still have a sentence, only now a wrong one.
A quantity is something you work out; a claim is something you check. In symbols the difference comes down to one mark: 4 \cdot 3 is a quantity, and 4 \cdot 3 = 12 is a claim. That mark sorts everything written in symbols into three kinds: expressions, equations and inequalities.
Definition
An expression is a way of writing a number. It is a single number, a single letter, or numbers and letters joined by operations: +, -, \cdot, \div, powers and brackets. So 4 \cdot 3, k + 11, 2^5, 3(c + 2), 40 and w are all expressions. Its pieces have names, the terms, coefficients and constants, and a letter in it stands for a number in the ways What a letter stands for describes. An expression has no verb. It says “this number” and nothing about that number.
An equation is two expressions joined by an equals sign, =. It claims that the two sides are the same number. Read aloud, the sign becomes a verb (“is equal to”, or just “is”), and the whole line becomes a sentence.
An inequality is two expressions joined by one of the order signs.
| Sign | Read as | Claims that the left side is |
|---|---|---|
| < | is less than | smaller than the right side |
| > | is greater than | larger than the right side |
| \le | is less than or equal to | smaller than the right side, or the same |
| \ge | is greater than or equal to | larger than the right side, or the same |
Equations and inequalities are claims, so each one is true or false. 3 + 4 = 7 is a true equation. 3 + 4 = 8 is also an equation, a false one: it makes a claim, and the claim is wrong. Being an equation is about how the line is built. Being true is about the numbers.
When a letter appears there is a third answer: it depends on the letter. n + 4 = 9 is true when n is 5 and false when n is any other number. n < 10 is true when n is 3 and false when n is 12. Until the letter has a number, the claim is waiting to be tested.
An expression is never true or false. “Is 4 \cdot 3 true?” has no answer, any more than “Is twelve true?” does.
Intuition
Think of the difference between a phrase and a sentence. “The heaviest pumpkin in the patch” is a phrase: it points at something and stops. “The heaviest pumpkin in the patch weighs 9 kilograms” is a sentence: it has a verb, and it tells you something you could check with a scale.
An expression is a phrase. An equation or an inequality is a sentence, and its sign is the verb. Figure 1 shows the shape every equation and every inequality has: an expression on each side, and the sign between them saying how the two are related.
A quick test is to put “Is it true that ...” in front. “Is it true that 3c + 2?” is not a question anyone can answer. “Is it true that 3c + 2 = 14?” is a real question, and the answer is “it depends on c”: yes when c is 4, since 3 \cdot 4 + 2 = 14, and no for any other number.
In words, a verb such as “is”, “costs” or “weighs” usually marks where a phrase stops and a claim begins.
Results
An expression has a value, not a truth value. Once every letter in it has a number, an expression works out to one number. It is never true or false.
An equation or an inequality has a truth value. With no letters, it is exactly one of true or false. With a letter, it may be true for some numbers and false for others.
A claim can be read from either end. a < b and b > a make the same claim. So do a \le b and b \ge a, and so do a = b and b = a.
“Or equal” means either one will do. a \le b is true when a < b is true or when a = b is true. So 5 \le 5 is true, while 5 < 5 is false.
Exactly one of three. For any two numbers a and b, exactly one of a < b, a = b and a > b is true.
Why it is true
Place numbers on a number line, with larger numbers further to the right. Then a < b means that a sits to the left of b, and a > b means that a sits to the right of b.
Two points on a line are either the same point, or one is further left than the other, and only one of those can happen: that is “exactly one of three”. The picture with a to the left of b is the same picture as b to the right of a, so a < b and b > a state one fact. The sign \le means “less than or equal to”, so it is true as soon as either part is.
An expression has no truth value because working it out gives a number, and a number claims nothing. Only a sign between two sides turns symbols into a sentence.
Worked examples
Sorting by the sign
Say whether each is an expression, an equation or an inequality.
(a) 5 \cdot 6 - 4 \qquad (b) 18 = 2 \cdot 9 \qquad (c) k + 11
(d) 7 < 3 + 5 \qquad (e) 2w \ge 30 \qquad (f) 40
SolutionLook for a sign that joins two sides. Nothing else decides the question.
(a) 5 \cdot 6 - 4 is an expression. There is no = and no order sign. It is a way of writing the number 26, and it says nothing about 26.
(b) 18 = 2 \cdot 9 is an equation: the expressions 18 and 2 \cdot 9 joined by =.
(c) k + 11 is an expression. It names the number that is 11 more than k.
(d) 7 < 3 + 5 is an inequality: 7 and 3 + 5 joined by <.
(e) 2w \ge 30 is an inequality: 2w and 30 joined by \ge.
(f) 40 is an expression, the simplest kind there is: a single number.
True or false
Is each claim true or false?
(a) 6 \cdot 4 = 24 \qquad (b) 15 - 6 = 8 \qquad (c) 2^3 > 3^2
(d) 12 \div 4 \le 3 \qquad (e) 5 \cdot 0 < 1
SolutionWork out each side, then compare the two numbers the way the sign says.
(a) 6 \cdot 4 is 24, so the claim is 24 = 24. It is true.
(b) 15 - 6 is 9, so the claim is 9 = 8, which is false. It is still an equation.
(c) 2^3 is 8 and 3^2 is 9, so the claim is 8 > 9. That is false, because 8 is less than 9.
(d) 12 \div 4 is 3, so the claim is 3 \le 3. “Less than or equal to” is true when either part holds, and 3 = 3, so the claim is true. With < in place of \le it would be false.
(e) 5 \cdot 0 is 0, so the claim is 0 < 1. It is true.
A claim that waits for its letter
Is p + 7 = 15 true when p = 8? When p = 10? Is 2q < 11 true when q = 3, when q = 5 and when q = 6?
SolutionPutting a number in place of the letter is substitution. It turns the claim into one with no letter, and that one is true or false.
For p + 7 = 15: when p = 8 the left side is 8 + 7, so the claim is 15 = 15, true. When p = 10 the left side is 10 + 7, so the claim is 17 = 15, false.
For 2q < 11: when q = 3 the claim is 6 < 11, true. When q = 5 it is 10 < 11, true. When q = 6 it is 12 < 11, false.
So neither claim is true or false on its own; each depends on its letter. A number that makes a claim true is called a solution of it.
Phrases and sentences at the ice cream van
The van’s board says: cone 3 dollars, tub 5 dollars. For each line, decide whether it is a phrase or a sentence, write it in symbols, and name what you wrote. If it has no letter, say whether it is true.
(a) the cost of c cones
(b) Cosmo buys four cones for 12 dollars.
(c) A tub costs more than a cone.
(d) the money Enzo has left after buying a tub, if he started with a dollars
(e) Enzo started with enough money for a tub.
Solution(a) There is no verb. The words name an amount, 3c dollars, so this is a phrase and 3c is an expression. Building it is translating words into an expression.
(b) The verb “buys ... for” claims that four cones cost 12 dollars: 4 \cdot 3 = 12. That is an equation, and since 4 \cdot 3 is 12, it is true.
(c) The verb “costs” compares two prices: 5 > 3. That is an inequality, and it is true.
(d) “The money Enzo has left” names an amount, a - 5 dollars. A phrase, so a - 5 is an expression.
(e) “Enough for a tub” means 5 dollars or more, so the claim is a \ge 5. The sign includes “equal”, because with exactly 5 dollars Enzo can still buy the tub. It is an inequality, and whether it is true depends on a.
More than, in a phrase and after a verb
Kasia’s pumpkin weighs w kilograms and Jago’s weighs j kilograms. Write each line in symbols and name what you wrote.
(a) six kilograms more than the weight of Kasia’s pumpkin
(b) Jago’s pumpkin weighs six kilograms more than Kasia’s.
(c) Jago’s pumpkin weighs more than six kilograms.
(d) Kasia’s pumpkin weighs no more than nine kilograms.
Solution(a) A phrase with no verb. “Six more than w” builds a number: w + 6, an expression. Here “more than” means add.
(b) The verb “weighs” joins Jago’s weight to the phrase from (a): j = w + 6, an equation. The phrase from (a) is still there, now as one side of the claim.
(c) This time “more than” comes straight after the verb and compares Jago’s weight with six: j > 6, an inequality. The same two words now give the sign >, not the operation +.
(d) “No more than nine” means nine or less: w \le 9, an inequality.
Inside a phrase “more than” builds a number; just after the verb it compares two numbers. If Kasia’s pumpkin weighs 5 kilograms and Jago’s weighs 11, then 11 = 5 + 6, 11 > 6 and 5 \le 9 are all true.
Reading a chain of equals signs
At the pumpkin patch, three pumpkins at 4 dollars each and a 2-dollar hayride cost 3 \cdot 4 + 2 dollars. Here are two ways of writing the working.
Which one makes only true claims?
SolutionEvery equals sign is a verb. A chain a = b = c claims that a = b and that b = c, and so that all three are the same number.
In (A) the first link claims 3 \cdot 4 + 2 = 12 + 2. Both sides are 14, so it is true. The second link, 12 + 2 = 14, is true too.
In (B) the first link claims 3 \cdot 4 = 12 + 2, which is the claim 12 = 14, and that is false. The second link is true, but the chain as a whole also claims 3 \cdot 4 = 14. It ends at the right cost while saying something false on the way. Only (A) makes nothing but true claims: it carries the “+\,2” along until it is used.
Choosing the sign
Which of =, <, >, \le and \ge make a true claim when put in the box?
(a) 2^4 \;\square\; 4^2 \qquad (b) 3^2 \;\square\; 2^3
(c) n + 3 \;\square\; n + 5, where n is any whole number
Solution(a) 2^4 is 16 and 4^2 is 16. The two sides are equal, so =, \le and \ge all make true claims, and < and > make false ones. More than one sign can be right.
(b) 3^2 is 9 and 2^3 is 8. The left side is larger, so > and \ge make true claims.
(c) Whatever whole number n is, n + 5 is 2 more than n + 3, so the left side is always the smaller. With < and \le the claim is true for every whole number n; with =, > and \ge it is false for every one. Here the answer happens to be the same for every number, and the letter is the placeholder in a general rule that What a letter stands for describes.
Boundaries
The sign decides, not the length. w on its own is an expression; x = 7 is an equation, short as it is.
A false equation is still an equation. Whether a line is an equation and whether it is true are separate questions.
Every equals sign makes a claim. In a chain of working, each = claims that the two things beside it are the same number, as Example 6 shows.
“It depends” also depends on which numbers are allowed. If p counts pumpkins, then p is a whole number, and 2p = 9 is false for every count, since half of 9 is not a whole number. The numbers a letter may stand for are part of the problem.
Not equal is a claim too. a \neq b, read “a is not equal to b”, is true or false in the same way. Books differ on whether to call it an inequality; either way it is a sentence.
Where this goes next
An expression is something to work out: put numbers in place of its letters and find its value. A claim with a letter is something to test, and the numbers that make it true are its solutions. Turning a story into such a claim is using a letter for an unknown.
An inequality is usually true for a whole stretch of numbers, which a number line shows at a glance. An equation true for every number, like n \cdot 1 = n, is an identity. Expressions with no letters are numerical expressions.
Exercises
Expression, equation or inequality
Name each one.
- 9 \cdot 7 + 2
- 45 = 5 \cdot 9
- h - 8
- 4r \le 36
- 13
- 20 > 2 \cdot 11
- b^2 + 1 = 26
True or false
- 7 + 6 = 14
- 48 \div 6 = 8
- 7^2 \ge 49
- 5^2 < 2^5
- 30 - 12 > 4 \cdot 5
Claims with a letter
- Is m + 9 = 17 true when m = 8? When m = 7?
- Is 3d > 20 true when d = 7? When d = 6?
- Is k \ge 5 true when k = 5? When k = 4?
Phrases and sentences
A scoop of ice cream from the van costs 2 dollars. Say whether each line is a phrase or a sentence, write it in symbols, and name what you wrote. If it has no letter, say whether it is true.
- the cost of s scoops of ice cream
- Enzo picked 7 pumpkins and Freja picked 4 more than Enzo, so Freja picked 11 pumpkins.
- four kilograms less than the weight of the biggest pumpkin, which weighs b kilograms
- The ice lolly Cosmo chose costs less than 4 dollars. (Call its price p dollars.)
- Five bags of 6 pumpkins hold 30 pumpkins.
Choosing the sign
Which of =, <, >, \le and \ge make a true claim when put in the box?
- 4^3 \;\square\; 8^2
- 2 \cdot 7 \;\square\; 3 \cdot 5
Reasoning
- Explain why 5 \cdot 8 + 1 cannot be called true or false, but 5 \cdot 8 + 1 = 41 can. Is the equation true?
- The working for “twice the sum of 6 and 9” is written 6 + 9 = 15 \cdot 2 = 30. Which equals sign makes a false claim? Rewrite the working so that every equals sign is true.
- Explain why n < n + 1 is true whatever whole number n stands for, while the truth of n > 4 depends on n. Give a number that makes n > 4 true and one that makes it false.
Answers
- An expression. There is no = and no order sign; it is a way of writing 65.
- An equation: 45 and 5 \cdot 9 joined by =.
- An expression: the number 8 less than h.
- An inequality: 4r and 36 joined by \le.
- An expression: a single number.
- An inequality: 20 and 2 \cdot 11 joined by >. It is false, since 2 \cdot 11 is 22, but that does not change its kind.
- An equation: b^2 + 1 and 26 joined by =.
- False. 7 + 6 is 13, and 13 = 14 is not so.
- True. 48 \div 6 is 8, and 8 = 8.
- True. 7^2 is 49, and 49 \ge 49 holds because the two sides are equal.
- True. 5^2 is 25 and 2^5 is 32, and 25 < 32.
- False. 30 - 12 is 18 and 4 \cdot 5 is 20, and 18 > 20 is not so.
- When m = 8: 8 + 9 is 17, so the claim is 17 = 17, true. When m = 7: 7 + 9 is 16, and 16 = 17 is false. Its truth depends on m.
- When d = 7: 3 \cdot 7 is 21, and 21 > 20 is true. When d = 6: 3 \cdot 6 is 18, and 18 > 20 is false.
- When k = 5: 5 \ge 5 is true, since the two sides are equal. When k = 4: 4 \ge 5 is false.
- A phrase, so an expression: 2s dollars.
- A sentence: 7 + 4 = 11, an equation, and it is true.
- A phrase, so an expression: b - 4 kilograms.
- A sentence: p < 4, an inequality. Whether it is true depends on p.
- A sentence: 5 \cdot 6 = 30, an equation, and it is true.
- 4^3 is 64 and 8^2 is 64. The sides are equal, so =, \le and \ge make true claims.
- 2 \cdot 7 is 14 and 3 \cdot 5 is 15. The left side is smaller, so < and \le make true claims.
- 5 \cdot 8 + 1 is an expression: it names a number and claims nothing. 5 \cdot 8 + 1 = 41 claims that this number is 41, and it is true: 5 \cdot 8 + 1 = 40 + 1 = 41.
- The first equals sign claims 6 + 9 = 15 \cdot 2, which is the claim 15 = 30, false. A version where every link is true is 2(6 + 9) = 2 \cdot 15 = 30.
- n + 1 is always one more than n, so n < n + 1 is true for every whole number. n > 4 compares n with a fixed number, so it holds for some numbers only: n = 5 makes it true, since 5 > 4, and n = 3 makes it false.