Inequalities
Some questions have a whole range of answers, and a number line is how you show them all.
"You must be at least 48 inches tall to ride."
Is that one height? No. Someone who is 48 can ride. So can someone who is 52, or 60.
Someone who is 46 cannot. The rule splits every possible height into two groups.
An equation would name one height. This names all of them at once.
That is an inequality: h ≥ 48. The answer is not a number but a stretch of the number line, and the dot at the end says whether 48 itself is included.
Reading the symbols
< is less than, > is greater than. The wide end points at the larger amount.
≤ and ≥ add "or equal to", which changes whether the endpoint itself counts.
"At least 48" is ≥ 48; "under 48" is < 48.
The line under the sign
That little line is the "or equal to" part. It decides whether 48 itself is allowed, which is the whole point of the ride rule.
Graphing on a number line
Mark the endpoint, then shade the direction that makes the statement true.
A filled circle means the endpoint is included; an open circle means it is not.
Shade right for greater than, left for less than — the direction the arrow would point.
Test a number
Pick any number in the shaded part and check it works. If it does not, you shaded the wrong way.
Everyday phrases
"No more than" is ≤. "At least" is ≥. "Fewer than" is <.
"Minimum" points the same way as "at least"; "maximum" the same way as "at most".
Reading the sentence carefully is most of the work — the symbol follows from the words.
Over and under exclude
"Over 21" does not include 21, so it is > 21 with an open circle. "21 and over" does include it.
Solving one-step inequalities
Solve exactly as you would an equation: do the same thing to both sides.
x + 3 ≤ 10 becomes x ≤ 7, and every number up to and including seven works.
The answer is a set, not a single value, so it is written as an inequality or drawn on a line.
One answer becomes many
An equation gives one number; an inequality gives a range. Checking means testing a value inside the range and one outside.
Does the answer make sense?
In a real problem, some solutions may be impossible even though the algebra allows them.
A number of people cannot be negative or a fraction, so n ≤ 7.5 really means at most seven people.
Always ask what the variable counts before reporting the answer.
Round the right way
With $50 and $7 tickets you can buy 7, not 7.14. For "at most" problems, always round down, whatever the decimal says.
Worked examples
Writing an inequality
Write "you must be at least 13 to join" as an inequality, using a for age.
- "At least" means the value counts. So the symbol includes equality.
- Thirteen-year-olds may join, and so may anyone older. The range goes upwards.
- a ≥ 13, drawn with a filled circle at 13 shading right. A twelve-year-old is excluded.
a ≥ 13
Solving
Solve x + 4 < 11 and describe the solutions.
- Subtract four from both sides. The same move as for an equation.
- x < 7. Strictly less — seven itself does not work.
- Every number below seven: 6, 0, -3, 6.9. An open circle at 7, shading left.
x < 7
A real limit
Tickets cost $7. With $50, how many can you buy?
- 7n ≤ 50, since you cannot spend more than you have. "At most" gives ≤.
- Divide both sides by seven: n ≤ 7.14. The algebra allows a fraction.
- But tickets are whole, so at most 7. Eight would cost $56, which is too much.
7 tickets
Words to know
- Inequality
- A statement that one amount is less than or greater than another.
- Solution set
- All the values that make the inequality true.
- At least
- Greater than or equal to, written ≥.
- At most
- Less than or equal to, written ≤.
- Open circle
- A hollow dot showing the endpoint is not included.
Mistakes to watch for
Shading the wrong way. For x < 7 the shading goes left, towards the smaller numbers. Testing one shaded value settles it in seconds.
Using a filled circle for a strict inequality. x < 7 excludes 7, so the circle must be open. x ≤ 7 includes it and gets a filled one.
Reading "over 21" as including 21. It does not — that is > 21. "21 and over" is the phrase that includes it.
Reporting a fractional answer to a counting problem. n ≤ 7.14 tickets means seven tickets; you cannot buy a seventh of one.
Rounding up on an "at most" problem. 7.14 rounds to 7 here, not 8, because 8 tickets would break the limit the inequality set.
The < and > signs were introduced by Thomas Harriot in a book published in 1631, ten years after he died.
Speed limits, age restrictions, weight limits and dosage ranges are all inequalities, and each one hinges on whether the endpoint counts.
The difference between "under 18" and "18 and under" has decided court cases — it is exactly the open-or-filled dot question.