Inequalities — Challenge

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

What these problems 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.

Problem 1

Which inequality says that x is at most 7?

  1. x≤ 7
  2. x<7
  3. x≥ 7
  4. x>7
  5. x=7
Worked answer

x≤ 7

At most means it can be 7 but no more than 7.

Because 7 itself is allowed, the sign includes equality: x≤ 7.

The strict sign x<7 would rule out 7, which at most does not.

Problem 2

Which of these numbers satisfies x>4?

  1. 0
  2. 2
  3. 3
  4. 5
  5. 4
Worked answer

5

The inequality asks for numbers strictly greater than 4.

Of the options only 5 is greater than 4.

4 itself does not count, because the sign is strict.

Problem 3

What is the greatest whole number x for which x+3<10?

  1. 5
  2. 6
  3. 7
  4. 13
  5. 3
Worked answer

6

Subtract 3 from both sides: x<7.

The sign is strict, so 7 is not allowed and the greatest whole number below it is 6.

Problem 4

Which inequality says that x is at least 12?

  1. x>12
  2. x≥ 12
  3. x≤ 12
  4. x<12
  5. x=12
Worked answer

x≥ 12

At least means 12 or anything larger.

Since 12 itself is allowed, the sign includes equality: x≥ 12.

Problem 5

Which of these numbers is not a solution of x≤ 8?

  1. 0
  2. 6
  3. 7
  4. 9
  5. 8
Worked answer

9

The inequality allows 8 and everything below it.

9 is greater than 8, so it is the one that fails.

Note that 8 is a solution, because the sign includes equality.

Problem 6

A ride is for children under 18. Which inequality describes an age a that is allowed?

  1. a<18
  2. a≤ 18
  3. a>18
  4. a≥ 18
  5. a=18
Worked answer

a<18

Under 18 means younger than 18, so an age of exactly 18 is not allowed.

That is the strict inequality a<18.

Writing a≤ 18 would admit an eighteen-year-old.

Problem 7

What is the greatest whole number x for which 2x≤ 14?

  1. 5
  2. 6
  3. 7
  4. 12
  5. 28
Worked answer

7

Divide both sides by 2: x≤ 7.

Equality is allowed, so 7 itself qualifies and is the greatest such whole number.

Problem 8

How many whole numbers satisfy x<5?

  1. 4
  2. 5
  3. 6
  4. 7
  5. 0
Worked answer

5

The whole numbers start at 0, so the solutions are 0,1,2,3,4.

Counting them gives 5 numbers. Forgetting 0 would give 4.

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