In an equation you can do almost anything to both sides and equal stays equal. An inequality has one trap: multiplying or dividing by a negative number. Then the larger side becomes the smaller. Once you have seen why, you stop getting it wrong.
1. What an inequality is
An inequality is a mathematical statement about the order of numbers, quantities or expressions. There are four symbols: < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to).
The signs have a history. Thomas Harriot (1560-1621) used < and >, and his work was published in 1631. ≤ and ≥ were first used by John Wallis (1616-1703) and Pierre Bouguer, in 1670 and 1734 respectively.
An equation is solved by a few fixed values. An inequality is solved by a range. x > 0 is true only for positive x. But x² > 0 is true for every real x except x = 0.
2. The same step on both sides
Addition: for a, b, c ∈ ℝ, a < b gives a + c < b + c, and a > b gives a + c > b + c. For example, 2 < 3 gives 2 + 1 < 3 + 1.
Subtraction: a < b gives a − c < b − c, and a > b gives a − c > b − c. For example, 4 > 3 gives 4 − 2 > 3 − 2.
Multiplication: if a < b and c > 0 then ac < bc; but if a < b and c < 0 then ac > bc. Multiplying by a negative reverses the inequality. 2 < 3 gives (2)(4) < (3)(4), yet (2)(−4) > (3)(−4), because −8 > −12.
Division works the same way: if a < b and c > 0 then a/c < b/c; if a < b and c < 0 then a/c > b/c. 2 < 3 gives 2/3 < 3/3, but 2/(−3) > 3/(−3).
Below, pick a < b, choose a step and drag c. The top line shows a and b, the bottom line shows where the step sent them.
both sides at oncedrag c
When does the order flip?
Now say it without the picture: does each step keep the direction or flip it?
pick the directionchip, then bin
Does the sign flip?
Pick up a chip, then drop it in a bin. Multiply or divide by a negative and it flips.
Right. Next one.
Not yet. First ask whether the multiplier or divisor is positive or negative.
All in place. −8 > −12, −2/3 > −1 and 1/2 > 1/3: those three flipped.
3. Proving the properties
Every proof turns on one key: a < b means b − a > 0. Transitivity: if a < b and b < c then a < c. Add b − a > 0 and c − b > 0 to get (b − a) + (c − b) > 0, that is c − a > 0. Add a to both sides: c > a. Put the steps in order yourself.
order the stepsfrom the first step
Proof of transitivity
Start from the first given fact.
Yes. What comes next?
Not yet. Both facts must be written before you add two positives, and c − a > 0 must come before you add a.
Right order. Two positives add to a positive, then add a.
If a < b then a + c < b + c. Proof: b − a > 0; adding c to both sides gives b − a + c > c, that is b + c > a + c.
If a < b and c > 0 then ac < bc. Proof: b − a > 0 and c > 0, and a product of two positives is positive, so (b − a)c > 0, that is bc − ac > 0. Adding ac gives bc > ac. If c < 0 then (b − a)c < 0, and the same steps give ac > bc.
If a < b and c < d then a + c < b + d. Proof: add a − b < 0 and c − d < 0 to get (a + c) − (b + d) < 0; adding (b + d) gives a + c < b + d.
If a < b with a > 0 and b > 0, then 1/a > 1/b. Proof: ab > 0, so 1/(ab) is positive. Multiplying a < b by 1/(ab) gives a/(ab) < b/(ab), that is 1/b < 1/a.
4. A property behind every step
Every step in solving an inequality is one of these four properties. From 3x − 2 > 2x − 1, subtract 2x to get x − 2 > −1, then add 2 to get x > 1. Each time, check whether the multiplier or divisor is positive or negative. Drop each step into the box of its property.
which property?chip, then bin
The reason for each step
Is the multiplier positive or negative? Pick up a chip, then drop it in a bin.
Right. Next one.
Not yet. Say first what was done to both sides.
All in place. Subtract 2x, add 2, multiply by 2 > 0, and divide by −3 < 0, which is why the last one flipped.
5. Check yourself
Three questions in the exam's style. The first is about the history of the signs.
check yourselfpick one answer
Question 1
According to the book, who first used ≤ and ≥ in 1670?
Thomas Harriot does not fit here; the right answer is John Wallis. The book says John Wallis used ≤ and ≥ in 1670; Bouguer followed in 1734.
Right. The book says John Wallis used ≤ and ≥ in 1670; Bouguer followed in 1734.
Pierre Bouguer does not fit here; the right answer is John Wallis. The book says John Wallis used ≤ and ≥ in 1670; Bouguer followed in 1734.
Georg Cantor does not fit here; the right answer is John Wallis. The book says John Wallis used ≤ and ≥ in 1670; Bouguer followed in 1734.
The next comes from inside a proof.
check yourselfpick one answer
Question 2
In proving ac < bc for c > 0, why is (b − a)c > 0?
because b − a < 0 does not fit here; the right answer is because the product of two positives is positive. a < b ⇒ b − a > 0 and c > 0; the product is positive (O₆).
Right. a < b ⇒ b − a > 0 and c > 0; the product is positive (O₆).
because c = 0 does not fit here; the right answer is because the product of two positives is positive. a < b ⇒ b − a > 0 and c > 0; the product is positive (O₆).
because a = b does not fit here; the right answer is because the product of two positives is positive. a < b ⇒ b − a > 0 and c > 0; the product is positive (O₆).
In the last, say which is always true.
check yourselfpick one answer
Question 3
If a < b, which of the following is always true?
Right. Adding the same number to both sides keeps the direction: a + 3 < b + 3.
a − 3 > b − 3 does not fit here; the right answer is a + 3 < b + 3. Adding the same number to both sides keeps the direction: a + 3 < b + 3.
3a > 3b does not fit here; the right answer is a + 3 < b + 3. Adding the same number to both sides keeps the direction: a + 3 < b + 3.
−a < −b does not fit here; the right answer is a + 3 < b + 3. Adding the same number to both sides keeps the direction: a + 3 < b + 3.
The whole note on eight cards.
cut from the notetap to turn over
Eight cards
QuestionWhat is the main difference between solving an equation and an inequality?
AnswerAn equation gives fixed values; an inequality gives a range of values.
QuestionFor which value of x is x² > 0 false?
Answerx = 0; it is true for every other real x.
QuestionIf a < b and c < 0, how are ac and bc related?
Answerac > bc (the inequality reverses).
QuestionIf a < b and c < 0, how are a/c and b/c related?
Answera/c > b/c.
QuestionIn the transitivity proof, what does adding b − a > 0 and c − b > 0 give?
Answerc − a > 0; then adding a gives c > a, i.e. a < c.
QuestionIf a < b and c < d, how are a + c and b + d related?
Answera + c < b + d.
QuestionIf 0 < a < b, how are 1/a and 1/b related?
Answer1/a > 1/b.
QuestionAccording to the book, who first used ≤ and ≥ and when?
AnswerJohn Wallis in 1670 and Pierre Bouguer in 1734 respectively.
Source: Higher Mathematics 2nd Paper, Classes 11-12 (NCTB), Chapter 1 “Real Numbers and Inequalities”, section 1.3.