The subtraction rule — to subtract a number, add its opposite — applied to single subtractions and to chains of them, the fact that subtraction is not commutative, evaluating a function containing a subtraction, identifying the terms of an expression written as a sum, and computing a change as a signed difference.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 2 — Properties of Real Numbers
Subtracting Real Numbers
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 86-92 — the lesson these objectives are drawn from
Warm-up
You met addition of negatives last lesson. This lesson shows that you have already learned subtraction too.
Discussion prompt
Work out 5 plus negative 3, then work out 5 minus 3. What do you notice, and could that be a coincidence?
Hint: Try a second pair, such as 2 plus negative 6 and 2 minus 6.
Answer:
\[ 5 + (-3) = 2 \qquad 5 - 3 = 2 \]
\[ 2 + (-6) = -8 \;? \quad \text{no} \quad 2 + (-6) = -4 \qquad 2 - 6 = -4 \]
Both pairs agree, and it is not a coincidence. Adding the opposite of a number and subtracting the number are the same operation written two ways, which means every subtraction you will ever meet can be turned into an addition you already know how to do.
Concept
Adding the opposite of a number is equivalent to subtracting the number. That turns subtraction from a second operation with its own rules into a rewriting step followed by the addition rules from Lesson 2.3.
subtraction rule — To subtract b from a, add the opposite of b to a. In symbols, a minus b equals a plus the opposite of b. The result is called the difference of a and b.
\[ a - b = a + (-b) \qquad \text{Example: } 3 - 5 = 3 + (-5) \]
After the rewrite there are no subtractions left, so only the two addition rules are ever needed.
Figure (svg): Two pairs of equivalent problems, one written as an addition of a negative and one as a subtraction
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 86-86
Section
Section 1
Concept
To subtract b from a, add the opposite of b to a. The rewrite changes two things at once: the operation sign becomes a plus, and the number's sign flips.
Both changes happen together. Making one without the other changes the value of the expression.
Figure (svg): The subtraction rule stating that a minus b equals a plus the opposite of b, with the example 3 minus 5
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 86-86 — the Subtraction Rule box
Picture it
The operation flips and the number flips, in the same move.
Figure (svg): A rewrite shown as two simultaneous changes: the operation sign flips and the number's sign flips
Four minus negative nine becomes four plus nine, which is thirteen. Two minus signs went in and none came out, which is exactly what the paired flip does.
Worked example
This is Example 1 from the textbook. The third one has a negative being subtracted.
\[ \text{Find } \; 10 - 11, \quad -11 - 10, \quad -4 - (-9). \]
Rewrite the first as an addition
Why: Add the opposite of eleven, which is negative eleven.
\[ 10 + (-11) \]
Apply the addition rules
Why: Opposite signs: eleven minus ten is one, with the sign of the larger, which is negative.
\[ -1 \]
Rewrite and evaluate the second
Why: Negative eleven plus negative ten: same sign, so add and keep the negative.
\[ -11 + (-10) = -21 \]
Rewrite and evaluate the third
Why: The opposite of negative nine is nine, so this becomes negative four plus nine.
\[ -4 + 9 = 5 \]
Figure (svg): The solution to Worked example three differences shown as a ladder of expressions, one row per algebraic move
\[ 10 - 11 = -1, \quad -11 - 10 = -21, \quad -4 - (-9) = 5 \]
Verify: check the third one against the number line
Why: Starting at negative four and adding nine moves nine units right, which crosses zero and lands at five. Subtracting a negative moved the answer up, which is what adding a positive always does — and that is the surprising case worth confirming.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 86-86
Translation
Four subtractions, four rewrites. Watch both signs.
Match the pairs
Why: In every rewrite the operation becomes an addition and the second number's sign flips. Notice the two where a negative was being subtracted: both turn into additions of positives, which is why subtracting a negative always increases the answer. The first number is never touched by the rewrite.
Worked example
Guided Practice 1 to 4. Rewrite every subtraction before computing anything.
\[ \text{Find } \; -3 - 5, \quad 12.7 - 10, \quad -1 - (-2), \quad 7 - 23. \]
Negative 3 minus 5 becomes negative 3 plus negative 5
Why: Same sign after the rewrite, so add and keep the negative.
\[ -8 \]
12.7 minus 10 becomes 12.7 plus negative 10
Why: Opposite signs: 12.7 minus 10 is 2.7, positive.
\[ 2.7 \]
Negative 1 minus negative 2 becomes negative 1 plus 2
Why: The opposite of negative two is two.
\[ 1 \]
7 minus 23 becomes 7 plus negative 23
Why: Opposite signs: 23 minus 7 is 16, with the negative sign of the larger.
\[ -16 \]
Figure (svg): The solution to Worked example four more from guided practice shown as a ladder of expressions, one row per algebraic move
\[ -8, \quad 2.7, \quad 1, \quad -16 \]
Verify: check which answers came out positive and why
Why: Only the two where the number being taken away was smaller in effect stayed positive. In the third, subtracting a negative acted as an addition and pushed the answer above the starting value — the only one of the four that increased.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 87-87
Trap
\[ -4 - (-9) \]
Change the subtraction to an addition and leave the negative nine as it is
Why: The rewrite is remembered as change the sign, and only one sign gets changed.
\[ -4 + (-9) = -13 \quad \text{(wrong)} \]
The correct answer is five. Flipping only the operation turned a subtraction of a negative into an addition of a negative, which is the opposite of what was intended.
\[ -4 - (-9) = -4 + 9 = 5 \]
Flip the operation and the number's sign together, as one move
Why: The rule says add the opposite: the word add supplies one flip and the word opposite supplies the other.
A quick check: subtracting a negative always makes the answer larger, because it is adding a positive. If your answer went down, only one flip happened.
Sorting
Rewrite each one mentally, then decide the direction relative to the first number.
Sort into buckets
Sort each subtraction by whether the answer is larger or smaller than the first number.
The direction depends entirely on the sign of the number being subtracted, never on the first number. Two of the falling cases start negative and two of the rising ones do too.
Faded example
Supply the operation and the sign.
Fill in the blanks
-4 - (-9) \;=\; -4 \; + \; 9 \;=\; 5
Why: Add the opposite: the operation becomes a plus and negative nine becomes nine. The result is negative four plus nine, which is five. Both blanks are filled by the same phrase in the rule, which is why the two changes have to be made together rather than one at a time.
Prediction
Commit before you compute.
Predict first
Is 6 minus negative 4 larger or smaller than 6?
Correct: Larger — it is 10.
\[ 6 - (-4) = 6 + 4 = 10 \]
Why: The rewrite gives six plus four, which is ten. Subtracting a negative is adding a positive, so the answer rises. This is the case that most contradicts the everyday meaning of the word subtract, and testing yourself on it deliberately is worth more than any amount of practice on the ordinary cases.
Section
Section 2
Concept
When an expression contains several subtractions, rewrite all of them as additions before doing any arithmetic. After that step the expression is a sum, and the addition rules and properties apply to the whole thing at once.
\[ -3 - (-4) - 12 = -3 + 4 + (-12) \]
Once everything is a sum, you may reorder and regroup freely, because addition is commutative and associative and subtraction is neither.
Figure (svg): The expression negative 5 minus x rewritten as a sum, with its two terms circled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 87-87 — Example 2, Expressions with More than One Subtraction
Picture it
Once it is a sum, every term carries its own sign.
Figure (svg): The expression negative 5 minus x rewritten as a sum, with its two terms circled
The circled pieces are the terms. Before the rewrite it is unclear whether the sign belongs to the number or to the operation; afterwards there is no ambiguity at all.
Worked example
Example 2 from the textbook. Rewrite both before touching anything.
\[ \text{Evaluate } \; -3 - (-4) - 12. \]
Rewrite both subtractions as additions
Why: Subtracting negative four becomes adding four; subtracting twelve becomes adding negative twelve.
\[ -3 + 4 + (-12) \]
Add the first two terms
Why: Opposite signs: four minus three is one, positive.
\[ 1 + (-12) \]
Add the last term
Why: Opposite signs: twelve minus one is eleven, with the negative sign of the larger.
\[ -11 \]
State the answer
Why: The whole expression comes to negative eleven.
\[ -11 \]
Figure (svg): The solution to Worked example an expression with two subtractions shown as a ladder of expressions, one row per algebraic move
\[ -3 - (-4) - 12 = -3 + 4 + (-12) = -11 \]
Verify: group by sign and re-add
Why: Positives total four; negatives total negative fifteen. Four plus negative fifteen is negative eleven, matching the left-to-right route. Two methods agreeing is a real check rather than a repetition.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 87-87
Ranking
Four moves, one correct sequence.
Put in order
Why: Counting the subtractions first tells you how many rewrites to make and gives you something to check against afterwards. Rewriting them all before any arithmetic is what makes the expression safe to reorder. Adding comes third, and the sign check comes last because it needs an answer to judge.
Worked example
Guided Practice 3 and 4 in the same style. The rewrite step is what keeps a long chain manageable.
\[ \text{Evaluate } \; -1 - (-2) - 6 \; \text{ and } \; 7 - 23 - 53. \]
Rewrite the first chain
Why: Subtracting negative two becomes adding two; subtracting six becomes adding negative six.
\[ -1 + 2 + (-6) \]
Total the first chain
Why: Negative one plus two is one; one plus negative six is negative five.
\[ -5 \]
Rewrite the second chain
Why: Both subtractions become additions of negatives.
\[ 7 + (-23) + (-53) \]
Total the second chain
Why: The two negatives total negative seventy-six; seven plus that is negative sixty-nine.
\[ -69 \]
Figure (svg): The solution to Worked example a longer chain shown as a ladder of expressions, one row per algebraic move
\[ -1 - (-2) - 6 = -5 \qquad 7 - 23 - 53 = -69 \]
Verify: check the sign of each answer against the terms
Why: In the first chain the negatives total seven against positives of two, so the answer must be negative. In the second the negatives total seventy-six against positives of seven, so it must be negative as well and by a much larger margin. Both signs match, and so do both magnitudes.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 87-87
Error analysis
The student evaluated two expressions containing subtractions. Both answers are wrong.
Annotate
On: \( -3 - (-4) - 12 = -3 - 4 - 12 = -19 \qquad 7 - 23 - 53 = 7 - (23 - 53) = 37 \)
Rewriting first prevents both errors at once: the first because the rewrite forces both flips, the second because a sum may be reordered while a chain of subtractions may not.
Elimination
The expression is negative 3 minus negative 4 minus 12.
Eliminate the wrong options
Which rewriting is right?
Survives elimination: A
Why: Each subtraction becomes an addition and each subtracted number flips sign, while the leading term is left alone. Negative four becomes positive four and twelve becomes negative twelve, giving negative eleven. The three wrong options each miss one half of the paired flip, which is the only way this rewrite ever goes wrong.
Fill the middle
The rewrite is done. Finish the addition.
Fill in the blanks
7 - 23 - 53 \;=\; 7 + (-23) + (-53) \;=\; 7 + (-76) \;=\; -69
Why: Grouping the two negatives gives negative seventy-six, and adding seven to that leaves negative sixty-nine. Once the chain is a sum the terms may be grouped in any order, which is exactly what makes the negatives collectable — a chain of subtractions offers no such freedom.
Socratic
You could evaluate a chain of subtractions left to right without rewriting anything.
Discussion prompt
Give two advantages of rewriting every subtraction as an addition before evaluating. At least one should be about a property you gain by doing so.
Hint: Think about what you are allowed to do with a sum that you are not allowed to do with a chain of subtractions.
Answer:
First, you gain the commutative and associative properties. A sum may be reordered and regrouped freely, so you can collect the positives and negatives into two piles and do one comparison at the end. A chain of subtractions must be worked strictly left to right.
Second, it removes the ambiguity about which sign belongs to which number. In a chain such as seven minus twenty-three minus fifty-three, the minus signs are operations; once rewritten they become part of the numbers, and each term is then a self-contained signed quantity that cannot be misread.
Section
Section 3
Concept
Addition may be performed in either order, but subtraction may not. Ten minus eleven is not the same as eleven minus ten — the two answers are opposites, so the order of the numbers affects the answer.
\[ 10 - 11 = -1 \qquad \text{but} \qquad 11 - 10 = 1 \]
This is exactly why the rewrite is worth doing: after it, the expression is a sum and the order stops mattering.
Figure (svg): Two columns showing that subtraction is not commutative, with 10 minus 11 and 11 minus 10 giving opposite answers
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 86-86 — the Study Tip on subtraction not being commutative
Picture it
One column takes the larger from the smaller; the other reverses it.
Figure (svg): Two columns showing that subtraction is not commutative, with 10 minus 11 and 11 minus 10 giving opposite answers
Negative one against positive one. Reversing a subtraction never gives a nearly-right answer — it gives the exact opposite, which makes the error easy to spot once you look for it.
Worked example
Example 1 parts a and b from the textbook, deliberately placed side by side.
\[ \text{Compare } \; 10 - 11 \; \text{ with } \; 11 - 10. \]
Rewrite and evaluate the first
Why: Ten plus negative eleven: opposite signs, eleven minus ten is one, sign of the larger is negative.
\[ 10 - 11 = -1 \]
Rewrite and evaluate the second
Why: Eleven plus negative ten: opposite signs, eleven minus ten is one, sign of the larger is positive.
\[ 11 - 10 = 1 \]
Compare the two answers
Why: They have the same size and opposite signs, so they are opposites.
\[ -1\text{ and } 1 \]
State the conclusion
Why: Subtraction is not commutative, since swapping the numbers changed the answer.
Figure (svg): The solution to Worked example compare both orders shown as a ladder of expressions, one row per algebraic move
\[ 10 - 11 = -1 \neq 1 = 11 - 10 \]
Verify: check that the two answers sum to zero
Why: Negative one plus one is zero, confirming they are opposites. That relationship holds for every reversed subtraction, which makes it a reliable check: if reversing a subtraction gave you anything other than the opposite, one of the two calculations is wrong.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 86-86
Sorting
For each pair of expressions, decide whether the two have the same value.
Sort into buckets
Sort each pair by whether the two expressions are equal.
The equal column and the unequal column contain the same numbers. What differs is whether the minus signs belong to the numbers or to the operations, and the rewrite is what moves them from the second category to the first.
Worked example
The rewrite is not just convenient — it buys back the properties subtraction lacks.
\[ \text{Show that } \; 7 - 3 \neq 3 - 7, \text{ but that the rewritten sums may be reordered.} \]
Evaluate both orders of the subtraction
Why: Seven minus three is four; three minus seven is negative four.
\[ 4\text{ and } -4 \]
Rewrite the first as a sum
Why: Seven plus negative three.
\[ 7 + (-3) \]
Reorder the sum
Why: Negative three plus seven, which is still four.
\[ -3 + 7 = 4 \]
State what the rewrite bought
Why: The sum may be reordered without changing its value; the subtraction may not.
Figure (svg): The solution to Worked example which properties survive the rewrite shown as a ladder of expressions, one row per algebraic move
\[ 7 - 3 = 7 + (-3) = -3 + 7 = 4 \qquad \text{but} \qquad 3 - 7 = -4 \]
Verify: check that the reordered sum is not the reversed subtraction
Why: Reordering seven plus negative three gives negative three plus seven, which keeps each number with its own sign and gives four. The reversed subtraction three minus seven moves the numbers without their signs and gives negative four. Those are two different operations, and confusing them is why the rewrite matters.
Trap
\[ 7 - 23 - 53 \]
Reorder to put the big numbers together, giving 23 - 53 - 7 or similar
Why: Reordering has been safe throughout Lesson 2.3, so the habit carries over.
Subtraction is not commutative, so moving the numbers without their signs changes the value. The reordered version gives negative thirty-seven rather than negative sixty-nine.
\[ 7 - 23 - 53 = 7 + (-23) + (-53) \]
Rewrite as a sum first, then reorder as much as you like
Why: Once every term carries its own sign, the terms may move freely because addition is commutative.
\[ = (-23) + (-53) + 7 = -76 + 7 = -69 \]
The rule is not that reordering is forbidden — it is that reordering must move each number together with its sign, and the rewrite is what attaches the signs.
Counterexample
One case is enough to refute a statement about every pair of numbers.
Discussion prompt
Someone claims that subtraction is commutative, meaning a minus b always equals b minus a. Give a counterexample, then say for which pairs of numbers the claim happens to be true.
Hint: Look for the pairs where the two answers coincide.
Answer:
\[ 10 - 11 = -1 \quad \text{but} \quad 11 - 10 = 1 \]
Ten and eleven refute the claim immediately. Reversing a subtraction always negates the answer, so the two sides agree only when a number equals its own opposite.
\[ a - b = b - a \;\Longleftrightarrow\; a = b \]
That happens exactly when the two numbers are equal, since then both sides are zero. So the claim holds only in the single degenerate case where there is nothing to swap — which is a good example of a claim that is technically true somewhere and useless everywhere.
Prediction
The relationship between the two answers is always the same.
Predict first
If a minus b equals 7, what is b minus a?
Correct: -7.
\[ b - a = -(a - b) = -7 \]
Why: Reversing a subtraction always negates the answer, so if one order gives seven the other gives negative seven. This holds for every pair of numbers, which makes it a useful check: compute a reversed subtraction and see whether you get the opposite of what you had.
Socratic
Lesson 2.3 listed five properties of addition. Subtraction does not inherit them all.
Discussion prompt
Subtraction fails the commutative property. Test whether it also fails the associative property, using the numbers 12, 5 and 3, and say what your result means for how a chain of subtractions must be evaluated.
Hint: Compare bracketing the first pair with bracketing the second pair.
Answer:
\[ (12 - 5) - 3 = 7 - 3 = 4 \qquad 12 - (5 - 3) = 12 - 2 = 10 \]
Subtraction fails the associative property too: the two bracketings give four and ten. So a chain of subtractions has no natural grouping and must be worked strictly from left to right, exactly as the left-to-right rule in Lesson 1.3 required.
This is the strongest argument for the rewrite. Turning every subtraction into an addition restores both properties at once, which is why the rewrite is worth doing even when the arithmetic would have been easy without it.
Section
Section 4
Concept
Evaluating a function containing a subtraction uses the routine from Lesson 1.8 with the rewrite inserted. Substitute the input, rewrite the subtraction as an addition, and apply the addition rules.
\[ y = 5 - x \quad \text{at } x = -2: \quad y = 5 - (-2) = 5 + 2 = 7 \]
When the input is negative, the substitution produces a subtraction of a negative — the case where the rewrite earns its keep.
Figure (svg): An input-output table for the function y equals 5 minus x at four values of x
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 87-87 — Example 3, Evaluate a Function
Picture it
Four inputs, two of them negative.
Figure (svg): An input-output table for the function y equals 5 minus x at four values of x
As the input rises by one the output falls by one, because the input is being subtracted rather than added. A falling step in the output row is the signature of a subtracted variable.
Worked example
Example 3 from the textbook, at x equal to negative 2, negative 1, 0 and 1.
\[ \text{Evaluate } y = 5 - x \text{ at } x = -2, -1, 0, 1 \text{ and organise the results.} \]
Substitute negative 2 and rewrite
Why: Five minus negative two becomes five plus two, which is seven.
\[ x = -2\text{ gives } y = 7 \]
Substitute negative 1 and rewrite
Why: Five minus negative one becomes five plus one, which is six.
\[ x = -1\text{ gives } y = 6 \]
Substitute 0
Why: Five minus zero is five; no rewrite is needed but it does no harm.
\[ x = 0\text{ gives } y = 5 \]
Substitute 1 and rewrite
Why: Five minus one becomes five plus negative one, which is four.
\[ x = 1\text{ gives } y = 4 \]
Figure (svg): An input-output table for the function y equals 5 minus x at four values of x
\[ y = 7, \; 6, \; 5, \; 4 \quad \text{for } x = -2, -1, 0, 1 \]
Verify: check the step between consecutive outputs
Why: Each output is one less than the one before, and each input is one more. A constant step of negative one is exactly what subtracting the variable should produce, and any irregular gap would point at a substitution error in that column.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 87-87
Faded example
The value is in place. Rewrite and evaluate.
Fill in the blanks
y = 5 - x \text+ x = -2: \quad y = 5 - (-2) = 5 7 2 = ___
Why: Subtracting negative two is adding two, so the operation becomes a plus and the answer is seven. The brackets round the substituted value are what keep the two minus signs from being merged, which is the only thing that can go wrong here.
Worked example
Guided Practice 5. Same shape, different constant and different inputs.
\[ \text{Evaluate } y = -4 - x \text{ at } x = -3, -1, 1, 3. \]
Substitute negative 3 and rewrite
Why: Negative four minus negative three becomes negative four plus three, which is negative one.
\[ x = -3\text{ gives } y = -1 \]
Substitute negative 1 and rewrite
Why: Negative four plus one is negative three.
\[ x = -1\text{ gives } y = -3 \]
Substitute 1 and rewrite
Why: Negative four plus negative one is negative five.
\[ x = 1\text{ gives } y = -5 \]
Substitute 3 and rewrite
Why: Negative four plus negative three is negative seven.
\[ x = 3\text{ gives } y = -7 \]
Figure (svg): The solution to Worked example tabulate y equals 4 minus x shown as a ladder of expressions, one row per algebraic move
\[ y = -1, \; -3, \; -5, \; -7 \quad \text{for } x = -3, -1, 1, 3 \]
Verify: check the step and the value at zero
Why: The inputs rise by two each time and the outputs fall by two each time, which matches a subtracted variable. And halfway between the second and third inputs, at x equal to zero, the rule gives negative four — the constant standing alone, exactly as the value at zero should.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 87-87
Trap
\[ y = 5 - x \text{ at } x = -2 \]
Substitute negative two by writing 5 - -2 and then reading the two minus signs as one
Why: Two adjacent signs look like a typing error, so one of them gets absorbed.
\[ 5 - 2 = 3 \quad \text{(wrong)} \]
The correct answer is seven. The two minus signs are doing different jobs — one is the operation and one belongs to the number — and merging them destroys one of them.
\[ y = 5 - (-2) = 5 + 2 = 7 \]
Put brackets round every negative value as you substitute it
Why: The brackets keep the number's sign visibly attached to the number, so the operation sign stays separate.
This is why Lesson 1.1 recommended substituting in brackets even when it seemed unnecessary. It becomes necessary the moment a negative value is substituted.
Pattern
Each frame adds one output to the table for y equals 5 minus x.
Step through it
The inputs rise and the outputs fall. What in the rule causes that, and what would change if the rule were y equals 5 plus x?
The variable is being subtracted, so every unit added to the input takes a unit off the output. With y equals 5 plus x the outputs would rise instead, giving 3, 4, 5, 6 — the same numbers read backwards.
Elimination
The function is y equals negative 4 minus x, evaluated at x equal to negative 3.
Eliminate the wrong options
What is the output?
Survives elimination: A
Why: Substituting gives negative four minus negative three, which rewrites to negative four plus three. Opposite signs, so subtract: four minus three is one, with the negative sign of the larger, giving negative one. Both flips of the rewrite are needed, and the leading negative four is untouched by them.
Socratic
The direction of the step tells you something about the rule.
Discussion prompt
A table's outputs fall by 3 each time its input rises by 1. What can you say about the rule, and how does that differ from a table whose outputs rise by 3? Give a rule of each kind.
Hint: Think about what has to be attached to the variable.
Answer:
A falling step means the variable is being subtracted, or equivalently multiplied by a negative. A step of negative three each time means three times the variable is being taken away, so the rule has the form y equals some constant minus 3x — for example y equals 10 minus 3x.
A rising step of three means three times the variable is being added, giving a rule such as y equals 10 plus 3x. The size of the step is what multiplies the variable and the direction of the step is its sign, which is exactly the reading Chapter 4 will formalise as the slope of a line.
Section
Section 5
Concept
When an expression is written as a sum, the parts that are added are called its terms. The subtraction rule is what lets you write any expression as a sum, and once you have, every term's sign is unambiguous.
term — One of the parts that are added when an expression is written as a sum. In negative five plus the opposite of x, the terms are negative five and the opposite of x.
\[ -5 - x = -5 + (-x) \quad \text{terms: } -5 \text{ and } -x \]
This matters for Lesson 2.7, where like terms are collected — and you cannot collect terms you cannot identify.
Figure (svg): The expression negative 5 minus x rewritten as a sum, with its two terms circled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 87-87 — the Terms of an Expression paragraph
Picture it
The rewrite makes the second term's sign explicit.
Figure (svg): The expression negative 5 minus x rewritten as a sum, with its two terms circled
Before the rewrite the minus sign could be read as an operation; afterwards it is part of the term. Lesson 2.7 relies entirely on that distinction.
Worked example
Write each expression as a sum first, then read off the terms.
\[ \text{Identify the terms of } \; -5 - x, \quad 3x - 7, \quad -2 - y + 4. \]
Rewrite the first as a sum
Why: Negative five plus the opposite of x.
\[ -5 + (-x) \]
Read off its terms
Why: Negative five and the opposite of x.
\[ -5\text{ and } -x \]
Rewrite and read the second
Why: Three x plus negative seven, so the terms are three x and negative seven.
\[ 3 x\text{ and } -7 \]
Rewrite and read the third
Why: Negative two plus the opposite of y plus four, giving three terms.
\[ -2, -y\text{ and } 4 \]
Figure (svg): The solution to Worked example identify the terms shown as a ladder of expressions, one row per algebraic move
\[ -5, -x \qquad 3x, -7 \qquad -2, -y, 4 \]
Verify: add the terms back and check the original
Why: Negative five plus the opposite of x is negative five minus x, which is what we started with. Reassembling the terms into the original expression confirms that no sign was lost or invented during the rewrite.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 87-87
Sorting
Rewrite each expression as a sum, then classify the term named.
Sort into buckets
Sort each term by its sign, once its expression is written as a sum.
Every minus sign in the original became part of a term after the rewrite. That is the whole content of the terms idea, and Lesson 2.7 depends on it to decide which terms may be combined.
Worked example
Example 5 in spirit. The change is today's closing price minus yesterday's.
\[ \text{A stock closes at } 48 \text{ then at } 45. \text{ Find the change.} \]
Write the change as a subtraction in the right order
Why: Change is the later value minus the earlier one, so today's close comes first.
\[ 45 - 48 \]
Rewrite as an addition
Why: Forty-five plus negative forty-eight.
\[ 45 + (-48) \]
Apply the addition rule
Why: Opposite signs: forty-eight minus forty-five is three, with the negative sign of the larger.
\[ -3 \]
Translate the sign back into words
Why: A negative change means the price fell.
\[ a\text{ fall of } 3\text{ dollars} \]
Figure (svg): Two closing prices with the change between them computed as a subtraction giving a negative result
\[ 45 - 48 = -3 \text{ dollars} \]
Verify: check the sign against the story
Why: The price went from forty-eight down to forty-five, so it fell, and the change came out negative. Had the subtraction been written the other way round the answer would have been positive three, which would have claimed a rise — so the order encodes the direction and has to be fixed by the meaning.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 88-88
Trap
\[ 48 - 45 = 3 \]
Take the smaller from the larger so that the answer comes out positive
Why: Years of arithmetic have made taking the small from the large automatic.
A change of positive three claims the stock rose. It fell, and the answer now says the opposite of the truth.
\[ 45 - 48 = -3 \]
Write the later value first, whatever its size
Why: The order is fixed by the definition of a change: the new value minus the old one. That is what makes a fall come out negative.
Subtraction being non-commutative is what makes this order carry meaning. If subtraction could be reversed freely, a change could not record a direction.
Translation
In each case, write the change as a subtraction in the correct order.
Match the pairs
Why: In every case the later value is written first. The third is worth studying: going from negative six up to two is a rise of eight degrees, and computing it requires subtracting a negative — which the rewrite turns into two plus six. Getting a rise of eight rather than a fall of four is entirely down to handling that double sign correctly.
Elimination
A temperature goes from negative 6 degrees to 2 degrees.
Eliminate the wrong options
Which expression gives the change?
Survives elimination: A
Why: A change is the later value minus the earlier one, so it is two minus negative six, which rewrites to two plus six and gives a rise of eight degrees. Checking the answer against the story settles it: the temperature went up, so the change must be positive, and only one option delivers that.
Socratic
The convention that a change is new minus old is not arbitrary.
Discussion prompt
Explain why defining a change as the later value minus the earlier one makes the sign of the answer meaningful. Then say what would go wrong if people were free to subtract in whichever order gave a positive answer.
Hint: Think about what information the sign is carrying.
Answer:
With the later value first, a rise gives a positive answer and a fall gives a negative one, so the sign records the direction of the change. That is only possible because subtraction is not commutative — the order carries information precisely because reversing it would change the answer.
If people subtracted in whichever order produced a positive result, every change would come out positive and the sign would carry no information at all. You would know how much something moved and never which way, which is exactly the difference between speed and velocity from Lesson 2.2 appearing again in a new setting.
Comparison
Fill the blanks from memory before you scroll back. The bottom row is what makes the rewrite worth doing.
Comparison matrix
| Addition | Subtraction | |
|---|---|---|
| Commutative? | Yes | No |
| Associative? | Yes | No |
| Can be rewritten as the other? | not needed | Yes, by adding the opposite |
Subtraction fails both rearrangement properties, and the rewrite converts it into an operation that has both. That is the entire argument for making the rewrite automatic.
Pattern
Whether the expression has one subtraction or five, the same five moves cover it.
Step two is a matched pair of changes. Every error in this lesson comes from making one of the two flips and not the other.
OpenStax Elementary Algebra 2e, §1.3 Add and Subtract Integers §1.3
Check
Subtracting a negative. Rewrite before you choose.
Check your understanding
What is negative 6 minus negative 10?
Answer: A
Why: The rewrite gives negative six plus ten. Opposite signs, so subtract: ten minus six is four, with the positive sign of the larger. Subtracting a negative raised the answer above the starting value, which is what it always does.
Check
A chain of subtractions. Rewrite them all first.
Check your understanding
Evaluate the expression 4 minus 9 minus negative 2.
Answer: A
Why: Rewriting gives four plus negative nine plus two. The negatives total nine and the positives total six, so the answer is negative three. Subtracting the negative two added two rather than removing it, which is what lifts the answer from negative five to negative three.
Check
A change. Fix the order from the meaning.
Check your understanding
A temperature falls from 3 degrees to negative 5 degrees. What is the change?
Answer: A
Why: The change is the later value minus the earlier one, so negative five minus three, which rewrites to negative five plus negative three and gives negative eight. The temperature fell, and a fall is recorded as a negative change.
Real world
A submarine is at negative 240 metres. Over the next hour it rises to negative 90 metres, then descends to negative 310 metres.
Discussion prompt
Compute each of the two changes as a signed difference, using the later value minus the earlier one, and say which required subtracting a negative. Then find the total change over the hour in two ways — by adding the two changes, and by comparing the final position with the starting one — and explain why the two agree.
Hint: Every position here is negative, so every change involves subtracting a negative.
Answer:
\[ \text{first change: } -90 - (-240) = -90 + 240 = 150 \text{ metres up} \]
\[ \text{second change: } -310 - (-90) = -310 + 90 = -220 \text{ metres, so down} \]
\[ \text{total by adding: } 150 + (-220) = -70 \qquad \text{total by comparing: } -310 - (-240) = -70 \]
Both changes required subtracting a negative, and both rewrites turned into additions of positives. The two routes to the total agree because the intermediate position cancels — going up 150 and then down 220 leaves you 70 below where you began, which is exactly what comparing the endpoints says.
That cancellation is the inverse property from Lesson 2.3 doing its work, and it is why changes can be added at all.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Is subtracting a negative number the same as adding a positive one?
Correct: Yes, always.
\[ -6 - (-10) = -6 + 10 = 4 \]
\[ 6 - (-10) = 6 + 10 = 16 \]
\[ 0 - (-10) = 0 + 10 = 10 \]
Why: The subtraction rule says a minus b equals a plus the opposite of b. When b is negative its opposite is positive, so subtracting it is adding a positive — and this holds for every value of the first number, positive, negative or zero. Negative six minus negative ten is negative six plus ten, and six minus negative ten is six plus ten. The first number plays no part in the rewrite at all.
Explain it
They can add signed numbers and find subtracting a negative baffling.
Discussion prompt
In no more than four sentences, explain why subtracting a negative number makes the answer bigger. Use an everyday situation rather than a rule, and then give them the two-part check they can run on any rewrite they perform.
Hint: Debt is the situation most people find convincing.
Answer:
A usable answer: think of a negative number as a debt. If you owe ten pounds and somebody removes that debt, you are ten pounds better off. Taking away something negative leaves you with more, not less, which is exactly what subtracting a negative does to a number.
The check has two parts, and both must happen together: the subtraction sign must become an addition sign, and the number after it must change sign. If they have made only one of those two changes, the answer will move the wrong way — and a quick glance at whether the answer went up or down catches it.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: The rewrite is fixed by making both flips as a single move and then checking whether the answer went the direction it should. Chains are fixed by rewriting every subtraction before any arithmetic, which also makes the terms reorderable. Substitution is fixed by putting brackets round every negative value as it goes in. Changes are fixed by always writing the later value first, whatever its size. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page write the subtraction rule in symbols and draw two arrows from it, one pointing at the operation sign and one at the number's sign, labelling both as things that flip together. Underneath, work one subtraction of a positive and one subtraction of a negative all the way through, and beside each write whether the answer ended up above or below the first number. In the middle, write one expression with at least three subtractions, rewrite it as a sum, and circle each term with its sign. Near the bottom, write the same two numbers as a subtraction both ways round and box the relationship between the two answers. Finally, in the margin, write a change from a real situation as a signed difference, with the later value first.
The boxed relationship should be that the two answers are opposites. If your two answers differ in any other way, one of the two subtractions has been computed incorrectly.
Recap
Five things, and the first one replaces an entire operation with one you already knew.
| If the question says | Your first move is |
|---|---|
| Find the difference | Rewrite as adding the opposite |
| Evaluate this chain | Rewrite every subtraction first |
| Evaluate the function at x = -2 | Substitute in brackets |
| Identify the terms | Write the expression as a sum |
| Find the change | Later value first, then subtract |
Lesson 2.5 turns to multiplication, where the sign question becomes simpler in one way and stranger in another: the answer's sign depends only on how many negative factors there are, and not on their sizes at all.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.4 Subtracting Real Numbers §2.4, pp. 86-92 — everything on these slides traces back here
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