Ordering numbers and interpreting inequalities
Use a common representation and a number line to compare values, including negatives and boundary points.
Section Paper 2 · Extended · Non-calculator
Core and Extended. The interval diagram also prepares you for algebraic inequalities. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Convert simple fractions and percentages to decimals.
- Locate zero and negative integers on a number line.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which is greater: −8 or −3?
What you will learn
- Order fractions, decimals and percentages without premature rounding.
- Compare negative numbers using their positions.
- Read strict and inclusive inequality symbols.
Right means greater
On a horizontal number line, values increase from left to right. Thus −5 < −2 even though 5 is larger than 2. Think of temperatures: −2°C is warmer than −5°C. Distance from zero measures magnitude in the sense of absolute size; position determines numerical order. A negative number farther from zero is smaller, not greater.
The symbol < means less than and > means greater than. The pointed end faces the smaller value: −5 < −2 can also be written −2 > −5. Read the statement aloud from left to right to check that it says what you intend.
Compare like with like
To compare 3/5, 0.58 and 62%, express them in one form: 0.60, 0.58 and 0.62. Then 0.58 < 3/5 < 62%. The decimal 0.60 equals 0.6; a zero at the end changes the written precision, not the value. By contrast, 0.06 is a different place value.
Fractions can also be compared using a common positive denominator. Compare 5/8 and 7/12 in twenty-fourths: 15/24 and 14/24, so 5/8 > 7/12. Comparing only the numerators would be misleading. If fractions are negative, keep the signs when converting; −15/24 < −14/24.
Use exact information when values are close
Rounding too early can hide an order. The exact fraction 2/3 is 0.6666… and is slightly less than the terminating decimal 0.667. Rounding both to two decimal places produces 0.67 and loses that distinction. Keep fractions exact or enough digits to establish the order.
To compare positive square roots, compare the numbers being rooted because the square-root function preserves order on non-negative inputs. For example, √5 > 2 because 5 > 4 = 2². This lesson uses that fact to estimate position; exact surd arithmetic belongs to the later surds topic.
A boundary may or may not be included
The symbols ≤ and ≥ allow equality; < and > do not. The statement −2 ≤ x < 3 means x is at least −2 but less than 3. On a number line, use a filled point at −2 and an open point at 3, with the interval between them marked. The notation ≠ means not equal; it does not specify which side a number lies on.
If x must be an integer, the permitted values are −2, −1, 0, 1 and 2. If x may be any real number, there are also infinitely many values between these integers, such as 0.5. Always read the permitted number type. In later algebra, multiplying an inequality by a negative number reverses its direction; here the number line explains why negatives require care.
Pause and explain
Does x = 3 satisfy −2 ≤ x < 3?
Worked example
Put −0.7, −2/3 and −68% in increasing order. Explain your method.
Show the worked solution
- Convert −68% to −0.68. Keep −2/3 as −0.6666… rather than rounding it prematurely.
- On the number line −0.700… lies left of −0.680…, which lies left of −0.6666….
- Increasing order runs from left to right, starting with the most negative value.
Answer −0.7 < −68% < −2/3.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Order 0.45, 2/5 and 43% from smallest to largest.
Give me a hint
Write all three as decimals first.
Compare my reasoning
- 2/5 = 0.40; 43% = 0.43.
- Compare hundredths: 40 < 43 < 45.
- Return to the original labels.
2/5 < 43% < 0.45.
Look for these in your work
- Used a common form.
- Wrote the original quantities in the requested order.
Order −3/4, −0.72 and −73%. Then list the integers satisfying −1 ≤ n < 3.
Give me a hint
Convert to −0.75, −0.72 and −0.73; the farther-left value comes first.
Compare my reasoning
- −0.75 < −0.73 < −0.72.
- Thus −3/4 < −73% < −0.72.
- The allowed integers are −1, 0, 1 and 2; 3 is excluded.
−3/4 < −73% < −0.72; −1, 0, 1, 2.
Look for these in your work
- Kept the direction correct for negatives.
- Respected each endpoint separately.
A sensor accepts values x with 2/3 ≤ x < 0.667. Will it accept 0.666, exactly 2/3, 0.6668 and 0.667? Explain without rounding the lower boundary.
Give me a hint
The exact lower boundary is 0.666666… . Compare each value to both boundaries.
Compare my reasoning
- 0.666 is below 2/3, so reject it.
- Exactly 2/3 is included by ≤, and 0.6668 lies between the two limits, so accept both.
- 0.667 equals the excluded upper boundary, so reject it.
Reject 0.666; accept 2/3 and 0.6668; reject 0.667.
Look for these in your work
- Preserved the exact recurring lower boundary.
- Checked both conditions for every candidate.
Common mistakes
- Comparing decimal strings by length instead of place value.
- Reversing the order of negative numbers.
- Including an endpoint marked with < or >.
- Replacing distinct exact values by equal rounded approximations.
What can you explain now?
List all integers n satisfying −3 < n ≤ 2.
Compare with the explanation
−2, −1, 0, 1, 2.
Exclude −3 because its boundary is strict. Include 2 because ≤ allows equality.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Tomorrow, compare −5/8 with −0.62 and explain the answer using both fractions and a number line.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.