Inequalities — Print

Some questions have a whole range of answers, and a number line is how you show them all.

What the practice questions cover

These questions use 1 idea from the course. Follow any of them for a definition, where it is taught, and what it leads to.

"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.

you must be at least 48 inches tall to ride 40" no 46" no 48" yes 56" yes 48" not one height — every height from 48 upwards

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.

Number line showing x is less than or equal to negative four -8-7-6-5-4-3-2-101234 x ≤ -4  ·  closed dot includes -4; shading runs left forever -4 is a solution an open circle would mean x < -4, leaving -4 out

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.

  1. "At least" means the value counts. So the symbol includes equality.
  2. Thirteen-year-olds may join, and so may anyone older. The range goes upwards.
  3. 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.

  1. Subtract four from both sides. The same move as for an equation.
  2. x < 7. Strictly less — seven itself does not work.
  3. 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?

  1. 7n ≤ 50, since you cannot spend more than you have. "At most" gives ≤.
  2. Divide both sides by seven: n ≤ 7.14. The algebra allows a fraction.
  3. 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.

Problem 1

Which symbol means "at least"?

  1. ≥
  2. >
  3. ≤
  4. <
Worked answer

≥

"At least" includes the value itself

Problem 2

Which symbol means "fewer than"?

  1. <
  2. ≤
  3. >
  4. ≥
Worked answer

<

Strictly less

Problem 3

Write "you must be at least 13 to join" using a for age. Use >= for the symbol.

Worked answer

a>=13

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Problem 4

Write "no more than 20 people" using p. Use <= for the symbol.

Worked answer

p<=20

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Problem 5

For x < 7, is the circle at 7 open or filled?

  1. Open
  2. Filled
  3. Either
  4. There is no circle
Worked answer

Open

7 < 7 is false, so 7 is excluded

Problem 6

Solve x + 4 < 11. Give the answer as x<number.

Worked answer

x<7

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Problem 7

Solve x - 3 ≥ 5. Give the answer as x>=number.

Worked answer

x>=8

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Problem 8

Solve 5x ≤ 40. Give the answer as x<=number.

Worked answer

x<=8

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Problem 9

Does "over 21" include 21?

  1. No
  2. Yes
  3. Sometimes
  4. Cannot tell
Worked answer

No

It means > 21, so 21 is excluded

Problem 10

Which value satisfies x ≥ -2?

  1. -1
  2. -5
  3. -3
  4. -10
Worked answer

-1

-1 ≥ -2 is true

Problem 11

How many whole numbers satisfy 1 ≤ n ≤ 6?

Worked answer

6

1,2,3,4,5,6

Problem 12

Tickets cost $7 and you have $50. How many can you buy?

Worked answer

7

n ≤ 7.14

Tickets are whole, so 7

Problem 13

A lift holds at most 600 kg. If each person is 75 kg, how many can ride?

Worked answer

8

n ≤ 8

Problem 14

A taxi charges $3 plus $2 a mile. With $20, how many whole miles can you go?

Worked answer

8

2m ≤ 17

m ≤ 8.5, so 8 whole miles

Problem 15

How many whole numbers satisfy -3 < n < 4?

Worked answer

6

-2,-1,0,1,2,3

Problem 16

Which phrase matches x ≤ 10?

  1. At most 10
  2. More than 10
  3. At least 10
  4. Fewer than 10
Worked answer

At most 10

"At most" is ≤

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