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Inequalities in words and symbols

Algebra · level 3, lesson 4 of 6

A question to start

A sign at a fair says: "To ride the roller coaster you must be at least 120 cm tall." Maya is 119 cm. Her brother is exactly 120 cm. Their cousin is 135 cm. Who can ride?

The brother and the cousin can ride; Maya cannot (not yet!). Notice something interesting: an equation like \(□ + 9 = 23\) has one answer, but the rule "at least 120 cm" has lots of answers: 120, 121, 135, 150, and every height above. This lesson is about sentences that have many answers, and the symbols for writing them.

The idea

Equal and not equal

An equation says two things are the same, like a balance scale that is level. But often two amounts are not the same: one is bigger. A sentence that says one amount is bigger or smaller than another is called an inequality ("not equal").

level: = tipped: left is more, >

The symbols < and >

  • \(7 > 3\) is read "7 is greater than 3".
  • \(3 < 7\) is read "3 is less than 7".

Both sentences say the same thing, looked at from opposite ends. How to remember which is which: the symbol is wide open on the side of the bigger number and comes to a point at the smaller number. The big end faces the big number. (Some people imagine a hungry mouth that always opens towards the bigger meal.)

"At least" and "at most"

Everyday words often mean "bigger or equal" or "smaller or equal".

Words Means Symbol
more than 5, greater than 5 6, 7, 8, ... (not 5) \(n > 5\)
less than 5, fewer than 5 4, 3, 2, ... (not 5) \(n < 5\)
at least 5, 5 or more, no fewer than 5 5, 6, 7, ... (5 is allowed) \(n \ge 5\)
at most 5, 5 or fewer, no more than 5 5, 4, 3, ... (5 is allowed) \(n \le 5\)

The symbols \(\ge\) and \(\le\) are the < and > signs with a line underneath. The line is half of an equal sign, a reminder that "equal" is allowed too. \(n \ge 5\) is read "\(n\) is greater than or equal to 5".

The whole difference between "more than 5" and "at least 5" is the number 5 itself. "More than 5" leaves 5 out; "at least 5" lets it in. That one number causes most mistakes with inequalities, so always ask: is the edge number allowed?

Many answers: the solution set

Look at the sentence \(n + 3 < 8\). Which whole numbers make it true? (The whole numbers are 0, 1, 2, 3, and so on, with no fractions.) Try them one at a time:

\(n\) \(n + 3\) Is it less than 8?
0 3 yes
1 4 yes
2 5 yes
3 6 yes
4 7 yes
5 8 no: 8 is not less than 8
6 9 no

So the answers are 0, 1, 2, 3 and 4. All the answers together are called the solution set. We can show them on a number line by putting a dot on each one:

0123 4567 89

A shortcut: find the edge first

Instead of testing every number, find the edge: the number that makes the two sides exactly equal. For \(n + 3 < 8\), the edge is where \(n + 3 = 8\), which is \(n = 5\). Now think: adding 3 to a smaller number gives a smaller answer. So every \(n\) below 5 gives a total below 8, and every \(n\) above 5 gives a total above 8. The edge itself gives exactly 8, which is not "less than 8", so 5 is out. The answer is "all whole numbers less than 5": 0, 1, 2, 3, 4.

Why can we trust this? Because \(n + 3\) always moves the same way as \(n\): if \(n\) goes up by 1, so does \(n + 3\). So the sentence switches from true to false at only one place, the edge.

Some solution sets go on forever

For \(2 \times n > 10\), the edge is \(n = 5\) (since \(2 \times 5 = 10\)), and 5 itself is not allowed. Bigger numbers give bigger doubles, so the answers are 6, 7, 8, 9, ... and they never stop. You cannot list them all, so you write "all whole numbers greater than 5", or \(n > 5\), or draw dots with an arrow going on.

Worked examples

1. Reading symbols. Write each in words and say if it is true: (a) \(12 > 9\); (b) \(4 \ge 4\); (c) \(15 < 10\).

(a) "12 is greater than 9": true. (b) "4 is greater than or equal to 4": true, because 4 is equal to 4. (c) "15 is less than 10": false.

2. Words to symbols. A lift can carry at most 8 people. Write this with \(p\) for the number of people, and list the possible numbers.

\(p \le 8\). Since you cannot have a fraction of a person, \(p\) can be 0, 1, 2, 3, 4, 5, 6, 7 or 8.

3. A solution set. List the whole numbers that make \(n + 6 \le 10\) true.

The edge: \(n + 6 = 10\) gives \(n = 4\). Here "equal" is allowed (\(\le\)), so 4 is in. Smaller \(n\) give smaller totals. The answers are 0, 1, 2, 3, 4. Check the edge: \(4 + 6 = 10\), and \(10 \le 10\) is true. ✓ Check 5: \(11 \le 10\) is false, so 5 is out. ✓

4. Between two numbers. A school trip needs at least 12 children and at most 40 (one bus). Write it with \(c\).

Both rules at once: \(12 \le c \le 40\), read "12 is less than or equal to \(c\), and \(c\) is less than or equal to 40". Any whole number from 12 to 40, including both ends, works.

5. Times inequalities. Which whole numbers make \(3 \times n < 20\) true?

Edge: \(3 \times 6 = 18\) and \(3 \times 7 = 21\), so 20 is between them. 6 gives 18 (less than 20, yes); 7 gives 21 (no). The answers are 0, 1, 2, 3, 4, 5, 6.

Try it

  1. Put < or > between each pair: (a) 45 and 54; (b) 0 and 3; (c) 1000 and 999.
  2. Write in symbols, using \(a\) for age: (a) you must be older than 10; (b) you must be at least 10; (c) children under 5 go free.
  3. List all the whole numbers that make \(n + 4 < 9\) true.
  4. List all the whole numbers that make \(n - 2 \le 3\) true. (Careful: \(n - 2\) needs \(n\) to be at least 2 here.)
  5. Describe all the whole numbers that make \(5 \times n > 30\) true. Can you list them all?
  6. A bag can hold at most 5 kg. You already have 2 kg in it. Write an inequality for \(w\), the weight you can still add, and give the largest whole number of kilograms.
  7. Is "at least 7" the same as "more than 7"? Give a number that shows the difference.
  8. Find a whole number that makes both \(n > 4\) and \(n < 6\) true. How many such numbers are there?
Answers
  1. (a) \(45 < 54\). (b) \(0 < 3\). (c) \(1000 > 999\).
  2. (a) \(a > 10\). (b) \(a \ge 10\). (c) \(a < 5\).
  3. Edge: \(n + 4 = 9\) at \(n = 5\), which is not allowed (< only). Answers: 0, 1, 2, 3, 4.
  4. Edge: \(n - 2 = 3\) at \(n = 5\), and \(\le\) lets 5 in. Smaller \(n\) give smaller results. Answers: 2, 3, 4, 5 (and if you allow subtracting past zero, 0 and 1 work too, since \(-2\) and \(-1\) are less than 3).
  5. Edge: \(5 \times 6 = 30\), which is not allowed. The answers are 7, 8, 9, 10, ..., every whole number greater than 6. They go on forever, so you cannot list them all; write \(n > 6\).
  6. \(2 + w \le 5\). The edge is \(w = 3\), which is allowed, so at most 3 kg more.
  7. No. 7 itself is "at least 7" but it is not "more than 7".
  8. \(n = 5\). It is the only one: 4 is not more than 4, and 6 is not less than 6.

Common mistake

Forgetting about the edge number. "Fewer than 5" does not include 5; "at most 5" does. Before writing your answer, test the edge number on its own and decide if it is in or out.

Mixing up < and >. Read the sentence aloud from left to right. \(3 < 7\) says "3 is less than 7". If what you say out loud is false, the symbol is the wrong way round.

Where it leads