Opposites as reflections across zero, absolute value as distance from zero and its three-case definition, solving simple absolute value equations and recognising when they have no solution, the difference between velocity and speed, and using a counterexample to disprove a general statement.
Subject: Algebra 1 · 65 slides · symbolic lesson
Open the interactive version of this deck
Title
Algebra 1 · Chapter 2 — Properties of Real Numbers
Absolute Value
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 2.2 Absolute Value §2.2, pp. 71-76 — the lesson these objectives are drawn from
Warm-up
You already treat distance as something that has no direction. This lesson gives that habit a symbol.
Discussion prompt
You walk 3 blocks east, then 3 blocks west. How far did you walk, and how far are you from where you started? Why are those two answers different?
Hint: One question is about the journey and one is about the position.
Answer:
You walked six blocks and you are zero blocks from home. Distance walked ignores direction and simply accumulates; displacement keeps the direction and lets the two legs cancel.
Absolute value is the operation that turns the second kind of quantity into the first. It throws away the sign and keeps the size, which is exactly what walking six blocks does to three east and three west.
Concept
The absolute value of a number is its distance from zero on a number line. Because a distance cannot be negative, an absolute value is never negative — whatever went in.
absolute value — The distance of a number from zero on a number line, written with a vertical bar on each side of the number.
Two numbers on opposite sides of zero at the same distance share an absolute value, and they are called opposites.
Figure (svg): Four numbers plotted on a number line with their distances from zero marked as absolute values
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 71-71
Section
Section 1
Concept
Two numbers that are the same distance from zero on a number line but on opposite sides of zero are opposites. Finding the opposite of a number means reflecting its point across zero.
opposite — One of two numbers the same distance from zero on a number line but on opposite sides of it. The opposite of a is written negative a.
The expression negative a can be read negative a or the opposite of a, and the second reading is the more useful one here.
Figure (svg): Negative 3 and 3 plotted on a number line, each three units from zero, joined by an arc labelled opposites
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 71-71 — the definition of opposites
Picture it
Each is three units from zero, and they sit on opposite sides.
Figure (svg): Negative 3 and 3 plotted on a number line, each three units from zero, joined by an arc labelled opposites
The arc joining them is the reflection. Everything about the two numbers is the same except which side of zero they are on, and that is exactly what being opposites means.
Worked example
This is Example 1 from the textbook. Do it on the line rather than by rule.
\[ \text{Use a number line to find the opposite of } -4. \]
Locate negative 4 on the line
Why: It is four units to the left of zero.
\[ -4\text{ is } 4\text{ units left} \]
Reflect across zero
Why: The opposite sits the same distance away, on the other side: four units to the right.
\[ 4\text{ units right} \]
Read the number at that point
Why: Four units to the right of zero is 4.
\[ \text{the opposite is } 4 \]
Note what the minus sign did
Why: Written as an operation, the opposite of negative four is negative negative four, which is four.
\[ -(-4) = 4 \]
Figure (svg): Negative 4 shown four units left of zero and 4 shown four units right, with an arrow showing the reflection
\[ -(-4) = 4 \]
Verify: check the distances match
Why: Negative four is four units from zero and four is also four units from zero, on opposite sides. Both conditions of the definition hold, so the two really are opposites.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 71-71
Matching
Four numbers, four opposites. Watch the signs.
Match the pairs
Why: Reflecting across zero changes the side and leaves the distance alone, so the digits never change and the sign always does. Zero is the exception that proves the rule: it sits exactly on the mirror, so its reflection is itself, and it is the only number that is its own opposite.
Worked example
Including the one case people hesitate over.
\[ \text{Find the opposite of } \; 7, \; -2.5, \; 0, \; \tfrac{1}{3}, \; -100. \]
The opposite of 7 is negative 7
Why: Reflect a point seven units right and it lands seven units left.
\[ -7 \]
The opposite of negative 2.5 is 2.5
Why: Reflecting a negative gives a positive; the minus signs cancel.
\[ 2.5 \]
The opposite of 0 is 0
Why: Zero is its own reflection, because it sits at the point everything is reflected across.
\[ 0 \]
The opposite of one third is negative one third
Why: The reflection works the same for fractions as for integers.
\[ -\frac{1}{3} \]
The opposite of negative 100 is 100
Why: The size never changes; only the side does.
\[ 100 \]
Figure (svg): The solution to Worked example opposites of five different numbers shown as a ladder of expressions, one row per algebraic move
\[ -7, \; 2.5, \; 0, \; -\tfrac{1}{3}, \; 100 \]
Verify: check that every pair adds to zero
Why: Seven and negative seven sum to zero; so do negative 2.5 and 2.5, and zero with itself. A number and its opposite always sum to zero, which is the algebraic version of being reflections of each other — and Lesson 2.6 will give that fact a name.
Trap
The opposite of a number is always negative.
Reason that the opposite is written with a minus sign, so it must be negative
Why: The notation negative a really does put a minus sign there, and the sign looks decisive.
\[ \text{The opposite of } -5 \text{ is } 5, \text{ which is positive.} \]
The claim fails for every negative number, which is half of all the numbers on the line.
The opposite of a number has the other sign, whatever the original sign was.
Read negative a as the opposite of a rather than as negative a
Why: The minus sign is an instruction to reflect, and reflecting a point on the left lands it on the right.
\[ -(5) = -5 \qquad -(-5) = 5 \qquad -(0) = 0 \]
This is Example 5 part a from the textbook, and it is the model counterexample for the whole chapter: one case is enough to refute always.
Sorting
For each number, decide the sign of its opposite.
Sort into buckets
Sort each number by the sign of its opposite.
Three columns for a question that looks binary. Zero needing its own column is a recurring theme in this chapter, and it is worth expecting rather than being surprised by.
Counterexample
A single case is enough to refute a statement that says always.
Discussion prompt
Someone claims that the opposite of a number is always negative. Give a counterexample, and then explain in one sentence why finding a second counterexample would add nothing.
Hint: You need one number whose opposite is not negative.
Answer:
\[ \text{the opposite of } -5 \text{ is } 5, \text{ which is positive} \]
Negative five is a counterexample, and one is enough because the claim asserted something about every number. A single failure makes always false, and a hundred failures make it no more false than one does.
This asymmetry between proving and disproving is worth holding on to: to establish a general claim you need an argument covering all cases, but to demolish one you need a single case. Example 5 in the textbook is built entirely on that asymmetry.
Prediction
Commit before you compute.
Predict first
What is the opposite of the opposite of 6?
Correct: 6.
\[ -(-(6)) = -(-6) = 6 \]
Why: The opposite of six is negative six, and the opposite of negative six is six again. Reflecting twice across the same mirror returns every point to where it started, which is why two minus signs in a row cancel. This is the same fact as negative negative six equalling six, seen as a geometric operation rather than a sign rule.
Section
Section 2
Concept
The absolute value of a number is its distance from zero, written with a vertical bar on each side. The definition splits into three cases according to the sign of the number inside.
\[ |3| = 3 \qquad |0| = 0 \qquad |-3| = -(-3) = 3 \]
The third line is the one worth staring at: when a is negative, negative a is positive.
Figure (svg): The three cases in the definition of absolute value, for a positive, zero, and a negative number
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 71-71 — The Absolute Value of a Number box
Picture it
Each row is a condition, a rule, and an example of that rule working.
Figure (svg): The three cases in the definition of absolute value, for a positive, zero, and a negative number
The three cases can be collapsed into one sentence: absolute value strips the sign and keeps the size. The three-case version is what you use when the number inside is a letter rather than a digit.
Worked example
This is Example 2 from the textbook. Two of the four have a minus sign outside the bars.
\[ \text{Evaluate } \; |5|, \quad |-2.3|, \quad -\left|\tfrac{1}{2}\right|, \quad -|-8|. \]
Five is positive, so its absolute value is itself
Why: First case of the definition, and the easiest.
\[ | 5 | = 5 \]
Negative 2.3 is negative, so take its opposite
Why: Third case: the absolute value is negative negative 2.3, which is 2.3.
\[ | - 2.3 | = 2.3 \]
For the third, work inside the bars first, then apply the outside minus
Why: The absolute value of one half is one half, and the minus in front makes the answer negative one half.
\[ -| \frac{1}{2} | = -\frac{1}{2} \]
For the fourth, evaluate inside the bars, then apply the outside minus
Why: The absolute value of negative eight is eight, and the minus in front gives negative eight.
\[ -| - 8 | = -8 \]
Figure (svg): The solution to Worked example evaluate four absolute value expressions shown as a ladder of expressions, one row per algebraic move
\[ |5| = 5, \quad |-2.3| = 2.3, \quad -\left|\tfrac{1}{2}\right| = -\tfrac{1}{2}, \quad -|-8| = -8 \]
Verify: check which answers came out negative and why
Why: Only the two expressions with a minus sign outside the bars produced negative answers, and the minus was applied after the absolute value had already made the inside positive. No absolute value itself was ever negative, which is what the definition guarantees.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 72-72
Sorting
For each expression, name the case of the definition that decides it.
Sort into buckets
Sort each expression by which case applies to the number inside the bars.
Half of these fall into the third case, which is where all the difficulty lives. The other two cases require you to change nothing at all.
Worked example
Guided Practice 1 to 4. Same routine, one case each.
\[ \text{Evaluate } \; |-4|, \quad |0|, \quad \left|\tfrac{3}{2}\right|, \quad |-1.7|. \]
Negative four is negative, so take its opposite
Why: Four units from zero, so the absolute value is four.
\[ | - 4 | = 4 \]
Zero is the middle case
Why: Zero is zero units from zero.
\[ | 0 | = 0 \]
Three halves is positive, so it is its own absolute value
Why: One and a half units from zero on the right.
\[ | \frac{3}{2} | = \frac{3}{2} \]
Negative 1.7 is negative, so take its opposite
Why: One point seven units from zero on the left.
\[ | - 1.7 | = 1.7 \]
Figure (svg): The solution to Worked example guided practice on absolute value shown as a ladder of expressions, one row per algebraic move
\[ 4, \quad 0, \quad \tfrac{3}{2}, \quad 1.7 \]
Verify: check that no answer is negative
Why: All four answers are zero or positive, which they must be, since every one of them is a distance. An absolute value that came out negative would be a signal that the outside of the bars had been confused with the inside.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 72-72
Error analysis
The student evaluated four absolute value expressions. Two are wrong.
Annotate
On: \( |5| = 5 \qquad |-2.3| = -2.3 \qquad -|-8| = 8 \qquad |0| = 0 \)
The two errors are mirror images: one lets a sign survive the bars that should not have, and one loses a sign outside the bars that should have survived. Working strictly inside-out prevents both.
Elimination
Three of these expressions have positive values.
Eliminate the wrong options
Which expression has a negative value?
Survives elimination: B
Why: The minus sign in the second option sits outside the bars, so it acts after the absolute value has already produced eight, giving negative eight. Every other option has all its signs inside the bars, where absolute value destroys them. Whether a minus sign is inside or outside the bars is the entire question.
Faded example
The inside is done. Apply what is left.
Fill in the blanks
-|-8| \;=\; -( 8 ) \;=\; -8
Why: The bars turn negative eight into its distance from zero, which is eight, and the minus sign waiting outside then negates that result. Working strictly from the inside out is what keeps the two signs from being confused, and it is the same discipline as working the innermost grouping first in Lesson 1.3.
Notation
This line looks wrong at first reading, and it is exactly right.
Annotate
On: \( \text{if } a < 0 \text{ then } |a| = -a \)
Reading negative a as the opposite of a rather than as a negative number is what makes this line readable. That is why the textbook introduces opposites before absolute value rather than after.
Section
Section 3
Concept
An equation such as the absolute value of x equals 7 asks a question: what numbers are 7 units from zero? Because a distance can be reached going either way, the answer is normally a pair.
\[ |x| = 7 \;\Longrightarrow\; x = 7 \text{ or } x = -7 \]
And because a distance is never negative, an equation setting an absolute value equal to a negative number has no solution at all.
Figure (svg): The equation the absolute value of x equals 7 shown as two points seven units from zero
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 72-72 — Example 3, Solve an Absolute Value Equation
Picture it
Two arrows of the same length, pointing in opposite directions.
Figure (svg): The equation the absolute value of x equals 7 shown as two points seven units from zero
The two arrowheads are the two solutions. Every absolute value equation with a positive right-hand side looks like this, which is why the answer comes in a pair rather than singly.
Worked example
This is Example 3 from the textbook. The third one is the interesting case.
\[ \text{Solve } \; |x| = 7, \quad |x| = 5.1, \quad |x| = -9. \]
Ask what numbers are 7 units from zero
Why: Both 7 and negative 7 are seven units away, so there are two solutions.
\[ x = 7\text{ or } x = -7 \]
Ask what numbers are 5.1 units from zero
Why: Both 5.1 and negative 5.1 are that far away.
\[ x = 5.1\text{ or } x = -5.1 \]
Ask what numbers are negative 9 units from zero
Why: A distance is never negative, so no point on the line satisfies this.
State each answer in the right form
Why: Two solutions, two solutions, and none at all.
Figure (svg): The equation the absolute value of x equals 7 shown as two points seven units from zero
\[ x = \pm 7, \quad x = \pm 5.1, \quad \text{no solution} \]
Verify: substitute each answer back
Why: The absolute value of 7 is 7 and the absolute value of negative 7 is also 7, so both check out. For the third equation there is nothing to substitute, and recognising that before starting is the whole skill — a distance set equal to a negative number is impossible on inspection.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 72-72
Sorting
Decide the number of solutions without solving anything.
Sort into buckets
Sort each equation by how many solutions it has.
Three possibilities, decided entirely by the sign of the right-hand side. Checking that sign first turns a solving problem into a reading problem.
Worked example
Guided Practice 5 to 7. One of the three has no solution.
\[ \text{Solve } \; |x| = 4, \quad |x| = 1.5, \quad |x| = -16. \]
Four units from zero
Why: Both 4 and negative 4 qualify.
\[ x = 4\text{ or } x = -4 \]
One point five units from zero
Why: Both 1.5 and negative 1.5 qualify.
\[ x = 1.5\text{ or } x = -1.5 \]
Negative sixteen units from zero
Why: Impossible: distance is never negative.
Notice the pattern in which ones failed
Why: The right-hand side being negative is what makes an equation impossible, and it can be spotted before any work is done.
Figure (svg): The equation the absolute value of x equals negative 9 shown as impossible, since distance cannot be negative
\[ x = \pm 4, \quad x = \pm 1.5, \quad \text{no solution} \]
Verify: count the solutions in each case
Why: Two, two, and none. An absolute value equation with a positive right-hand side has exactly two solutions, one with zero on the right has exactly one, and one with a negative on the right has none. Those three cases exhaust the possibilities.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 72-72
Trap
\[ |x| = 7 \]
Answer x equals 7 and stop
Why: Seven is the number visible in the equation, and it does satisfy it, so the work feels finished.
Negative seven is also seven units from zero and is equally a solution. Half the answer has been thrown away, and on a marked paper half the marks with it.
\[ |x| = 7 \;\Longrightarrow\; x = 7 \text{ or } x = -7 \]
Read the equation as a distance question and remember there are two directions
Why: Asking what numbers are seven units from zero makes the pair obvious in a way that solving does not.
A useful habit: whenever you see absolute value bars around the unknown, expect two answers, then check whether the right-hand side rules them out.
Prediction
The sign of the right-hand side settles this before any work.
Predict first
How many solutions does the equation the absolute value of x equals negative 3 have?
Correct: None.
\[ |x| \geq 0 \text{ for every } x, \text{ so } |x| = -3 \text{ is impossible} \]
Why: An absolute value is a distance from zero, and no distance is negative, so no value of x can make the left side equal negative three. Recognising this from the sign alone saves the effort of solving, and it is worth doing as the first step of every absolute value equation you meet.
Elimination
The equation is the absolute value of x equals 5.
Eliminate the wrong options
Which answer is complete and correct?
Survives elimination: A
Why: Both five and negative five are exactly five units from zero, so both satisfy the equation and a complete answer names both. Options B and C are each half of the right answer, and on a marked paper a half answer normally scores a half mark rather than a full one.
Socratic
Most equations so far have had exactly one solution. This kind usually has two.
Discussion prompt
Explain, using the number line, why an absolute value equation normally has two solutions while an equation such as x plus 3 equals 10 has only one. Then say what has to happen for the absolute value equation to have only one.
Hint: Think about how many ways there are to be a given distance away.
Answer:
An equation such as x plus 3 equals 10 fixes the position of x directly: there is exactly one point on the line where adding three gives ten. An absolute value equation fixes only the distance, and a distance can be achieved in two directions — left or right — so two points qualify.
The two collapse into one only when the distance is zero, since there is just one point zero units from zero. And if the required distance is negative there are no points at all. So the three cases are decided entirely by whether the right-hand side is positive, zero or negative, which is a much simpler question than solving.
Section
Section 4
Concept
Velocity tells you how fast something is moving and in what direction, so it can be positive or negative. Speed tells you only how fast, so it can only be positive. Speed is the absolute value of velocity.
\[ \text{speed} = |\text{velocity}| \]
By convention, up is positive and down is negative, so a falling object has a negative velocity and a positive speed.
Figure (svg): Two columns contrasting velocity, which carries a sign, with speed, which is its absolute value
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 73-73 — the Velocity and Speed paragraph and Example 4
Picture it
The left column keeps the direction; the right column discards it.
Figure (svg): Two columns contrasting velocity, which carries a sign, with speed, which is its absolute value
This is the clearest everyday example of absolute value doing real work: it converts a quantity that carries direction into one that does not, which is exactly what the word speed means.
Worked example
Example 4 from the textbook. The elevator drops at about 12 feet per second.
\[ \text{Find the velocity and the speed of an elevator dropping at } 12 \text{ feet per second.} \]
Decide the sign from the direction
Why: The motion is downward, and down is negative by convention.
State the velocity with its sign
Why: Negative twelve feet per second.
\[ \text{velocity } = -12 \text{ft} / s \]
Take the absolute value to get the speed
Why: The absolute value of negative twelve is twelve.
\[ | - 12 | = 12 \]
State the speed with its unit
Why: Twelve feet per second, with no direction attached.
\[ \text{speed } = 12 \text{ft} / s \]
Figure (svg): The solution to Worked example the launch pad elevator shown as a ladder of expressions, one row per algebraic move
\[ \text{velocity} = -12 \text{ ft/s}, \qquad \text{speed} = |-12| = 12 \text{ ft/s} \]
Verify: check that the speed is not negative
Why: Speed is never negative because it is an absolute value, and twelve is positive. If a speed ever came out negative, an absolute value was skipped somewhere.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 73-73
Matching
Up is positive and down is negative.
Match the pairs
Why: Downward motions take a negative velocity and upward ones a positive velocity, while every speed is positive or zero. The rising balloon is the only case where the two numbers agree exactly, and the object at rest is the only case where the speed is zero — which is the middle case of the absolute value definition appearing in physics.
Worked example
Guided Practice 8 and 9. A parachutist descends at about 17 feet per second.
\[ \text{Find the parachutist's velocity and speed.} \]
Identify the direction
Why: Descending means moving downward, which is the negative direction.
Write the velocity
Why: Negative seventeen feet per second.
\[ \text{velocity } = -17 \text{ft} / s \]
Take the absolute value
Why: The absolute value of negative seventeen is seventeen.
\[ | - 17 | = 17 \]
State the speed
Why: Seventeen feet per second.
\[ \text{speed } = 17 \text{ft} / s \]
Figure (svg): The solution to Worked example the parachutist shown as a ladder of expressions, one row per algebraic move
\[ \text{velocity} = -17 \text{ ft/s}, \qquad \text{speed} = 17 \text{ ft/s} \]
Verify: compare with the elevator
Why: The parachutist has the larger speed, seventeen against twelve, and the more negative velocity, negative seventeen against negative twelve. Faster downward motion means a larger speed and a smaller velocity at the same time, which is a good check that the two quantities are being kept apart.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 73-73
Trap
\[ \text{velocity} = -12 \text{ ft/s} \;\rightarrow\; \text{speed} = -12 \text{ ft/s} \]
Copy the velocity across and call it the speed
Why: The two quantities share a number and a unit, so they look interchangeable.
Nothing moves at negative twelve feet per second. Speed is how fast, and how fast has no direction to be negative in.
\[ \text{speed} = |-12| = 12 \text{ ft/s} \]
Apply the absolute value, which is what converts one quantity into the other
Why: The two quantities differ by exactly that operation, so skipping it means reporting the wrong quantity.
A quick check: any answer to a question about speed that carries a minus sign is wrong, whatever the arithmetic before it looked like.
Discrimination
Decide which quantity each statement is describing.
Sort into buckets
Sort each statement by the quantity it reports.
Estimation
The parachutist descends at about 17 feet per second.
Predict first
Roughly how far does the parachutist fall in 10 seconds, and what sign does that displacement carry?
Correct: About 170 feet downward, so about -170 feet.
\[ \text{displacement} = -17 \cdot 10 = -170 \text{ feet} \qquad \text{distance} = |-170| = 170 \text{ feet} \]
Why: Ten seconds at seventeen feet per second is about a hundred and seventy feet, and the motion is downward so the displacement is negative. The distance fallen is 170 feet and the displacement is negative 170 feet — the same relationship between speed and velocity, one level up.
Socratic
Speed and velocity give the same number for most everyday purposes.
Discussion prompt
Describe a situation where knowing only the speed of something would be inadequate and the velocity would be needed. Then say what mathematical operation converts the more informative quantity into the less informative one, and whether the reverse conversion is possible.
Hint: Think about two objects moving at the same speed in a way that matters.
Answer:
Two lifts in the same shaft both moving at fifteen feet per second is a safe situation if they are moving the same way and a collision if they are not. Speed alone cannot distinguish those cases; velocity can, because the sign records the direction.
Absolute value converts velocity into speed, discarding the direction. The reverse conversion is impossible: from a speed of fifteen you cannot recover whether the velocity was fifteen or negative fifteen. Absolute value throws information away permanently, which is exactly why an absolute value equation has two solutions rather than one.
Section
Section 5
Concept
To prove that a statement is true you must show it holds for all examples. To prove that a statement is false it is enough to show that it fails for a single example, which is called a counterexample.
counterexample — A single example showing that a general statement is false. One counterexample is enough; a hundred add nothing.
The asymmetry is the point: proving needs everything, disproving needs one thing.
Figure (svg): The three cases in the definition of absolute value, for a positive, zero, and a negative number
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 73-73 — the Counterexample paragraph and Example 5
Picture it
Most of the claims in this section are about these three lines.
Figure (svg): The three cases in the definition of absolute value, for a positive, zero, and a negative number
A claim about absolute value is true if it follows from all three cases and false if any one case breaks it. The negative case is where almost every false claim fails.
Worked example
Example 5 from the textbook. One claim is false and one is true.
\[ \text{(a) The opposite of a number is always negative. (b) The absolute value of a number is never negative.} \]
Test claim (a) on a positive number
Why: The opposite of 5 is negative 5, which is negative, so this case supports the claim.
\[ 5\text{ supports it} \]
Test claim (a) on a negative number
Why: The opposite of negative 5 is 5, which is positive.
\[ -5\text{ breaks it} \]
Declare claim (a) false and name the counterexample
Why: One failing case is enough, and negative 5 is it.
Test claim (b) against the definition
Why: All three cases of the definition give a result of zero or more, so the claim holds everywhere.
Figure (svg): The solution to Worked example true or false, with a counterexample shown as a ladder of expressions, one row per algebraic move
\[ \text{(a) false: } -(-5) = 5 \qquad \text{(b) true by definition} \]
Verify: check that the counterexample really is a counterexample
Why: The claim said always negative, and negative five gives an opposite of positive five. That single case contradicts the word always, so the claim is refuted — no further cases need to be examined at all.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 73-73
Sorting
Judge each claim, and for the false ones think of the case that breaks it.
Sort into buckets
Sort each claim by whether it is true.
Every false claim here breaks on a negative input, and every true one survives all three cases. When testing a claim in this chapter, try a negative number first — that is where the failures live.
Worked example
Guided Practice 10 to 12. Two of the three are false.
\[ \text{(10) } -a \text{ is never positive. (11) } |a| \text{ is always at least } a. \text{ (12) } |\text{negative number}| \text{ is always negative.} \]
Test claim 10
Why: If a is negative, say negative 3, then negative a is 3, which is positive.
Test claim 11 on a positive number
Why: If a is 5, the absolute value is 5, which equals a. Equal counts as at least.
\[ \text{holds for } a = 5 \]
Test claim 11 on a negative number
Why: If a is negative 5, the absolute value is 5, which is greater than negative 5.
\[ \text{holds for } a = -5 \]
Test claim 12
Why: The absolute value of negative 3 is 3, which is positive.
Figure (svg): The solution to Worked example three more claims to judge shown as a ladder of expressions, one row per algebraic move
\[ \text{10 false } (a = -3), \quad \text{11 true}, \quad \text{12 false } (a = -3) \]
Verify: check claim 11 in the case that decides it
Why: Claim 11 says the absolute value is at least a, and the only way it could fail is if the absolute value were smaller than a. For positive a the two are equal, and for negative a the absolute value is positive while a is negative, so it is strictly larger. Both cases hold, and zero gives equality, so the claim is true everywhere.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 73-73
Trap
The opposite of a number is always negative.
Try 5, get negative 5; try 8, get negative 8; try 100, get negative 100; conclude the claim is true
Why: Three confirmations feel like strong evidence, and they were obtained honestly.
All three test cases were positive numbers. The claim was never tested where it could fail, so the evidence is worthless.
Test the claim deliberately where it is most likely to break.
Ask which kind of number the claim is least comfortable with, and try that first
Why: For a claim about opposites, that is a negative number; for a claim about absolute value, also a negative number.
\[ -(-5) = 5 \quad \text{— the claim is false} \]
Looking for confirmation finds it almost every time. Looking for a failure is the only search that can settle the question.
Counterexample
A claim that is true for most numbers is still false if it fails for one.
Discussion prompt
Someone claims that the absolute value of a number is always greater than the number itself. Find a counterexample, and then say precisely how the claim should be reworded to make it true.
Hint: Look for a case where the absolute value and the number are the same.
Answer:
\[ |5| = 5, \text{ which is not greater than } 5 \]
Any positive number is a counterexample, and so is zero, since the absolute value equals the number itself in every one of those cases. The word greater is what breaks the claim.
The corrected version replaces greater with at least, or equivalently greater than or equal to. That is Guided Practice 11 from the textbook, and it is true for every real number: strictly greater for negatives, and exactly equal for zero and the positives.
Elimination
The claim is that the opposite of a number is always negative.
Eliminate the wrong options
Which of these is a valid counterexample?
Survives elimination: A
Why: A counterexample must be a specific case where the claim's conclusion fails. The opposite of negative five is positive five, which is not negative, so the word always is refuted. Option D is worth noting: stating that counterexamples exist is not the same as giving one, and only the latter settles anything.
Socratic
One case demolishes a claim; a thousand cases do not establish one.
Discussion prompt
Explain why a single counterexample refutes a statement containing the word always, but a thousand confirming examples do not prove it. Then describe what would be needed to actually prove a claim about all real numbers.
Hint: Count how many real numbers there are, and how many you could test.
Answer:
A claim saying always is a claim about every number, and there are infinitely many. Testing a thousand of them leaves infinitely many untested, so confirmation can never be complete. But the claim asserts there are no exceptions, so producing one exception contradicts it directly and finishes the argument.
To prove such a claim you need an argument that covers all cases at once rather than a list of instances. For absolute value that means arguing from the three-case definition: show the claim holds when the number is positive, when it is zero, and when it is negative, and those three cases exhaust every real number. That is exactly how claim (b) in Example 5 is settled — by definition rather than by testing.
Comparison
Fill the blanks from memory before you scroll back. The two operations are easy to confuse and behave quite differently.
Comparison matrix
| Opposite of a | Absolute value of a | |
|---|---|---|
| What it does on the line | reflects across zero | measures distance from zero |
| Applied to 5 | -5 | 5 |
| Applied to -5 | 5 | 5 |
| Can the answer be negative? | Yes | No, never |
The two agree on negative inputs and disagree on positive ones. That partial overlap is exactly why they get confused, and why the bottom row is the fastest way to tell them apart.
Pattern
Whether the question asks for an opposite, an absolute value, a solution or a judgement, the same five moves cover it.
Step two is what separates the absolute value of negative eight from the negative of the absolute value of negative eight — two expressions with the same symbols and opposite answers.
OpenStax Elementary Algebra 2e, §1.3 Add and Subtract Integers §1.3
Check
Absolute value with a sign outside. Work inside out.
Check your understanding
What is the value of the expression negative the absolute value of negative 6?
Answer: A
Why: Inside the bars, negative six has an absolute value of six. The minus sign outside the bars then negates that result, giving negative six. The bars are the innermost grouping, so they are completed before anything outside them acts.
Check
Solving. Read it as a distance question.
Check your understanding
Solve the equation the absolute value of x equals 9.
Answer: A
Why: The equation asks which numbers are nine units from zero, and there are two: nine on the right and negative nine on the left. Both check, since the absolute value of each is nine.
Check
Judging a claim. Test it where it is most likely to fail.
Check your understanding
Which statement is FALSE?
Answer: A
Why: The opposite of negative five is positive five, so the claim fails for every negative number — which is half of all the numbers there are. One counterexample is enough to refute a statement containing the word always.
Real world
A machine part is specified to be 50 millimetres long, and the tolerance allows the actual length to differ from that by at most 0.2 millimetres. An inspector measures a part at 49.75 millimetres.
Discussion prompt
Write the tolerance rule using absolute value, with L for the measured length, and decide whether this part passes. Then say what the two boundary lengths are, and why absolute value is the natural tool for a specification of this kind.
Hint: The phrase differ from by at most is a distance statement.
Answer:
\[ |L - 50| \leq 0.2 \]
\[ |49.75 - 50| = |-0.25| = 0.25 > 0.2 \;\rightarrow\; \text{the part fails} \]
The part is 0.25 millimetres away from the target and only 0.2 is permitted, so it is rejected — narrowly, and in the short direction. The two boundary lengths are 49.8 and 50.2 millimetres, exactly 0.2 away on each side.
Absolute value is natural here because the specification does not care which way the error goes. A part 0.25 too long and one 0.25 too short are equally out of tolerance, and absolute value is precisely the operation that treats those two cases identically. Chapter 6 turns this kind of statement into an inequality you can solve.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Can the expression negative a ever be a positive number?
Correct: Yes, whenever a is itself negative.
\[ a = -3 \;\Longrightarrow\; -a = -(-3) = 3 > 0 \]
\[ a = 3 \;\Longrightarrow\; -a = -3 < 0 \]
Why: The expression negative a means the opposite of a, and the opposite of a negative number is positive. With a equal to negative three, negative a is negative negative three, which is three. This is exactly why the third case of the absolute value definition can read as negative a and still deliver a positive answer — and why reading negative a as a negative number is the single most damaging misreading in this chapter.
Explain it
They can plot numbers on a line and have never seen the vertical bars.
Discussion prompt
In no more than four sentences, explain what the bars around a number mean, without using the word absolute. Then tell them why an equation with bars usually has two answers, and give one everyday quantity that behaves the way an absolute value does.
Hint: Start from a question about the number line rather than from a rule.
Answer:
A usable answer: the bars ask how far the number is from zero, ignoring which side it is on. Negative seven and seven are both seven steps from zero, so the bars give seven for both. Distance has no direction, which is why the answer is never negative.
An equation with bars asks which numbers are a given distance from zero, and you can go that far in two directions, so there are usually two answers. Speed is the everyday version: a lift going down at twelve feet per second and one going up at twelve feet per second have the same speed, because speed records how fast without recording which way.
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: Opposites are fixed by reading the minus sign as reflect rather than as make negative. Signs outside the bars are fixed by working strictly inside out, one line at a time. Missing solutions are fixed by reading the equation as a distance question and expecting a pair before you start. Counterexamples are fixed by testing every claim on a negative number first, since that is where the claims in this lesson break. 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
Draw a number line across the top of a page and mark a pair of opposites on it, joining them with an arc labelled with the distance they share. Underneath, write the three cases of the absolute value definition, and beside each case write one worked example of it. In the middle of the page, write one absolute value equation with two solutions, one with a single solution and one with none, and beside each write the sign of its right-hand side. Near the bottom, write the two words velocity and speed with one motion described under each, showing the operation that converts the first into the second. Finally, in the margin, write one false claim about absolute value and the single counterexample that kills it.
Your three equations should have right-hand sides that are positive, zero and negative respectively. If two of them have the same sign, one of the three cases is missing.
Recap
Five things, and the second one is the one that has two signs to keep straight rather than one.
| If the question says | Your first move is |
|---|---|
| Find the opposite | Reflect the point across zero |
| Evaluate the absolute value | Check the sign of what is inside the bars |
| Solve for x | Check the sign of the right-hand side |
| Find the speed | Take the absolute value of the velocity |
| True or false | Try a negative number first |
Lesson 2.3 starts using the negative half of the line rather than merely describing it: adding real numbers, where the rules depend on whether the two signs agree, and where absolute value turns out to be exactly the tool for stating those rules.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.2 Absolute Value §2.2, pp. 71-76 — everything on these slides traces back here
Want this taught 1-on-1? Alexander tutors Algebra 1 — $55/session, free consultation.