GCSE Maths: Inequality symbols (Higher)
Part of the GCSE Maths inequalities chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Inequality symbols
An equation says that two things are exactly equal, and uses the sign =. But many situations in real life are not exact. A fairground ride says “You must be at least 120 cm tall”. A lift says “Maximum 8 people”. A phone plan gives you “up to 5 GB” of data. None of these tells you one exact number. They tell you what is allowed: a whole range of numbers. In maths we write these ideas with inequality symbols.
There are four symbols. Each one compares two numbers or quantities.
| Symbol | Read as | Example |
|---|---|---|
| < | is less than | 3 < 7 (3 is smaller than 7) |
| > | is greater than | 9 > 2 (9 is bigger than 2) |
| ≤ | is less than or equal to | 5 ≤ 5 and 4 ≤ 5 are both true |
| ≥ | is greater than or equal to | 8 ≥ 8 and 9 ≥ 8 are both true |
How to remember which way round. Think of the symbol as a hungry mouth. The wide open end always faces the bigger number, and the small pointed end points at the smaller number. So 3 < 7 and 7 > 3 say exactly the same thing.
The line underneath. The extra line under ≤ and ≥ adds the words “or equal to”. So x ≥ 4 means “x is 4 or more”, and 4 itself is allowed. But x > 4 means “x is more than 4”, and 4 itself is not allowed.
Inequalities with a letter. When the inequality contains a letter, such as x > 3, it does not have one answer. It describes every number that makes it true: 3.1, 4, 7.5, 100 and so on. We call any such number a solution. To test a number, replace the letter with it and see whether the statement is true.
Negative numbers. On a number line, numbers get bigger as you go right and smaller as you go left. So −5 is smaller than −2, even though 5 looks bigger than 2. We write −5 < −2. A temperature of −5°C is colder than −2°C.
Turning words into symbols. These phrases appear in exam questions all the time. Learn which have the “or equal to” line.
| Words | Symbol |
|---|---|
| more than, greater than, over, above, exceeds | > |
| less than, fewer than, under, below | < |
| at least, no less than, a minimum of | ≥ |
| at most, no more than, a maximum of | ≤ |
Key ideas
- < means “less than” and > means “greater than”. The open end of the symbol faces the bigger number.
- ≤ and ≥ include the number itself (“or equal to”). < and > do not.
- A solution of an inequality is any number that makes it true. There are usually many solutions.
- “At least” means ≥ and “at most” means ≤.
- With negative numbers, the number further to the left is smaller: −7 < −3.
Key words. An inequality symbol tells you which of two numbers is bigger. < means “is less than” and > means “is greater than”.
Method. Put both numbers on a number line. The one further to the right is bigger. The open end of < or > faces the bigger number.
- Picture both numbers on a number line. Numbers get bigger as you move to the right and smaller as you move to the left.−6 and −2 on the same number line
- Decide which one is further to the right. The number further right is the bigger one. −2 is further right.−6 is to the left of −2
- Choose the symbol. The symbol is like a hungry mouth: the open end always faces the bigger number.−6 < −2
Check. Read it back in words: “−6 is less than −2”. Both numbers are negative: the one nearer to zero is the bigger one. ✓
Watch out. With negative numbers, a number that looks “bigger” is actually smaller: −5 < −2 because −5 is further to the left.
Key words. At least means this or more. At most means this or less.
Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.
- Find the key phrase. The phrase tells us which symbol to use.“at least”
- Decide if the number itself is allowed. “at least” allows the boundary number itself, so we need the “or equal to” line under the symbol.≥
- Write it with the letter on the left. a stands for the age of the rider in years.a ≥ 12
Check. Read it back: “a is greater than or equal to 12” means the same as the sentence. ✓
Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).
Key words. A solution is a number that makes the inequality true.
Method. Put each number in place of the letter and decide whether the inequality is true.
- Write the inequality and what it means. We test each number by replacing x with it.x ≤ 5, which says “x is less than or equal to 5”
- Test every number in turn. A number is a solution only if the inequality is true for it. Compare using a number line if you need to.−3 ≤ 5? True ✓ allowed
5 ≤ 5? True ✓ allowed
5.5 ≤ 5? False ✗ not allowed
4.9 ≤ 5? True ✓ allowed
0 ≤ 5? True ✓ allowed - List the numbers that work. Keep only the ones marked true.−3, 5, 4.9, 0
Check. The boundary is 5. Numbers to the left of it work, and the boundary itself works, which agrees with the list. ✓
Watch out. Check the boundary number 5 carefully. With ≤ it is a solution.
Key words. A statement is true if it is correct and false if it is wrong.
Method. Say the statement in words, compare the numbers on a number line, then decide.
- Read the statement in words. Say what the symbol means.−4 ≥ −4 reads “−4 is bigger than or equal to −4”
- Compare the two numbers. Use a number line if you need to: further right means bigger.−4 is equal to −4
- Decide whether the statement is true. For ≤ and ≥ the statement is true if the numbers are equal or the inequality holds.True
Check. Test the opposite claim: −4 < −4 would be false, which agrees. ✓
Watch out. ≤ and ≥ are true when the two numbers are equal as well. So 5 ≤ 5 is true.
Watch out
“At least 12” means 12 or more, so it is a ≥ 12, not a > 12. The word “at” is the clue that the number itself is allowed. Also be careful with negative numbers: −9 is smaller than −3.Exam tip
If you are unsure which way a symbol points, put the two numbers on a number line. The bigger number is on the right, and the open end of the symbol faces it.Exercise 10A Reading and writing inequalities
Non-calculatorKey words. An inequality symbol tells you which of two numbers is bigger. < means “is less than” and > means “is greater than”.
Method. Put both numbers on a number line. The one further to the right is bigger. The open end of < or > faces the bigger number.
- Picture both numbers on a number line. Numbers get bigger as you move to the right and smaller as you move to the left.7 and 3 on the same number line
- Decide which one is further to the right. The number further right is the bigger one. 7 is further right.7 is to the right of 3
- Choose the symbol. The symbol is like a hungry mouth: the open end always faces the bigger number.7 > 3
Check. Read it back in words: “7 is greater than 3”. That makes sense. ✓
Watch out. The symbol must point towards the smaller number, with the wide open end at the bigger one.
Key words. An inequality symbol tells you which of two numbers is bigger. < means “is less than” and > means “is greater than”.
Method. Put both numbers on a number line. The one further to the right is bigger. The open end of < or > faces the bigger number.
- Picture both numbers on a number line. Numbers get bigger as you move to the right and smaller as you move to the left.2 and 9 on the same number line
- Decide which one is further to the right. The number further right is the bigger one. 9 is further right.2 is to the left of 9
- Choose the symbol. The symbol is like a hungry mouth: the open end always faces the bigger number.2 < 9
Check. Read it back in words: “2 is less than 9”. That makes sense. ✓
Watch out. The symbol must point towards the smaller number, with the wide open end at the bigger one.
Key words. An inequality symbol tells you which of two numbers is bigger. < means “is less than” and > means “is greater than”.
Method. Put both numbers on a number line. The one further to the right is bigger. The open end of < or > faces the bigger number.
- Picture both numbers on a number line. Numbers get bigger as you move to the right and smaller as you move to the left.−4 and 1 on the same number line
- Decide which one is further to the right. The number further right is the bigger one. 1 is further right.−4 is to the left of 1
- Choose the symbol. The symbol is like a hungry mouth: the open end always faces the bigger number.−4 < 1
Check. Read it back in words: “−4 is less than 1”. That makes sense. ✓
Watch out. The symbol must point towards the smaller number, with the wide open end at the bigger one.
Key words. An inequality symbol tells you which of two numbers is bigger. < means “is less than” and > means “is greater than”.
Method. Put both numbers on a number line. The one further to the right is bigger. The open end of < or > faces the bigger number.
- Picture both numbers on a number line. Numbers get bigger as you move to the right and smaller as you move to the left.−3 and −8 on the same number line
- Decide which one is further to the right. The number further right is the bigger one. −3 is further right.−3 is to the right of −8
- Choose the symbol. The symbol is like a hungry mouth: the open end always faces the bigger number.−3 > −8
Check. Read it back in words: “−3 is greater than −8”. Both numbers are negative: the one nearer to zero is the bigger one. ✓
Watch out. With negative numbers, a number that looks “bigger” is actually smaller: −5 < −2 because −5 is further to the left.
Key words. An inequality symbol tells you which of two numbers is bigger. < means “is less than” and > means “is greater than”.
Method. Put both numbers on a number line. The one further to the right is bigger. The open end of < or > faces the bigger number.
- Picture both numbers on a number line. Numbers get bigger as you move to the right and smaller as you move to the left.0.5 and 0.25 on the same number line
- Decide which one is further to the right. The number further right is the bigger one. 0.5 is further right.0.5 is to the right of 0.25
- Choose the symbol. The symbol is like a hungry mouth: the open end always faces the bigger number.0.5 > 0.25
Check. Read it back in words: “0.5 is greater than 0.25”. That makes sense. ✓
Watch out. The symbol must point towards the smaller number, with the wide open end at the bigger one.
Key words. An inequality symbol tells you which of two numbers is bigger. < means “is less than” and > means “is greater than”.
Method. Put both numbers on a number line. The one further to the right is bigger. The open end of < or > faces the bigger number.
- Picture both numbers on a number line. Numbers get bigger as you move to the right and smaller as you move to the left.−1.5 and −1.2 on the same number line
- Decide which one is further to the right. The number further right is the bigger one. −1.2 is further right.−1.5 is to the left of −1.2
- Choose the symbol. The symbol is like a hungry mouth: the open end always faces the bigger number.−1.5 < −1.2
Check. Read it back in words: “−1.5 is less than −1.2”. Both numbers are negative: the one nearer to zero is the bigger one. ✓
Watch out. With negative numbers, a number that looks “bigger” is actually smaller: −5 < −2 because −5 is further to the left.
Key words. A statement is true if it is correct and false if it is wrong.
Method. Say the statement in words, compare the numbers on a number line, then decide.
- Read the statement in words. Say what the symbol means.6 > 2 reads “6 is bigger than 2”
- Compare the two numbers. Use a number line if you need to: further right means bigger.6 is bigger than 2
- Decide whether the statement is true. The statement is true only if the numbers really are in that order.True
Check. Test the opposite claim: 6 < 2 would be false, which agrees. ✓
Key words. A statement is true if it is correct and false if it is wrong.
Method. Say the statement in words, compare the numbers on a number line, then decide.
- Read the statement in words. Say what the symbol means.−5 > −2 reads “−5 is bigger than −2”
- Compare the two numbers. Use a number line if you need to: further right means bigger.−5 is smaller than −2
- Decide whether the statement is true. The statement is true only if the numbers really are in that order.False
Check. Test the opposite claim: −5 < −2 would be true, which agrees. ✓
Key words. A statement is true if it is correct and false if it is wrong.
Method. Say the statement in words, compare the numbers on a number line, then decide.
- Read the statement in words. Say what the symbol means.4 ≤ 4 reads “4 is smaller than or equal to 4”
- Compare the two numbers. Use a number line if you need to: further right means bigger.4 is equal to 4
- Decide whether the statement is true. For ≤ and ≥ the statement is true if the numbers are equal or the inequality holds.True
Check. Test the opposite claim: 4 > 4 would be false, which agrees. ✓
Watch out. ≤ and ≥ are true when the two numbers are equal as well. So 5 ≤ 5 is true.
Key words. A statement is true if it is correct and false if it is wrong.
Method. Say the statement in words, compare the numbers on a number line, then decide.
- Read the statement in words. Say what the symbol means.9 ≥ 10 reads “9 is bigger than or equal to 10”
- Compare the two numbers. Use a number line if you need to: further right means bigger.9 is smaller than 10
- Decide whether the statement is true. For ≤ and ≥ the statement is true if the numbers are equal or the inequality holds.False
Check. Test the opposite claim: 9 < 10 would be true, which agrees. ✓
Watch out. ≤ and ≥ are true when the two numbers are equal as well. So 5 ≤ 5 is true.
Key words. A statement is true if it is correct and false if it is wrong.
Method. Say the statement in words, compare the numbers on a number line, then decide.
- Read the statement in words. Say what the symbol means.−3 < −1 reads “−3 is smaller than −1”
- Compare the two numbers. Use a number line if you need to: further right means bigger.−3 is smaller than −1
- Decide whether the statement is true. The statement is true only if the numbers really are in that order.True
Check. Test the opposite claim: −3 > −1 would be false, which agrees. ✓
Key words. A statement is true if it is correct and false if it is wrong.
Method. Say the statement in words, compare the numbers on a number line, then decide.
- Read the statement in words. Say what the symbol means.0 ≥ −7 reads “0 is bigger than or equal to −7”
- Compare the two numbers. Use a number line if you need to: further right means bigger.0 is bigger than −7
- Decide whether the statement is true. For ≤ and ≥ the statement is true if the numbers are equal or the inequality holds.True
Check. Test the opposite claim: 0 < −7 would be false, which agrees. ✓
Watch out. ≤ and ≥ are true when the two numbers are equal as well. So 5 ≤ 5 is true.
Key words. At least means this or more. At most means this or less.
Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.
- Find the key phrase. The phrase tells us which symbol to use.“at most”
- Decide if the number itself is allowed. “at most” allows the boundary number itself, so we need the “or equal to” line under the symbol.≤
- Write it with the letter on the left. p stands for the number of people in the lift.p ≤ 8
Check. Read it back: “p is less than or equal to 8” means the same as the sentence. ✓
Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).
Key words. At least means this or more. At most means this or less.
Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.
- Find the key phrase. The phrase tells us which symbol to use.“at least”
- Decide if the number itself is allowed. “at least” allows the boundary number itself, so we need the “or equal to” line under the symbol.≥
- Write it with the letter on the left. a stands for the age in years.a ≥ 14
Check. Read it back: “a is greater than or equal to 14” means the same as the sentence. ✓
Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).
Key words. At least means this or more. At most means this or less.
Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.
- Find the key phrase. The phrase tells us which symbol to use.“less than”
- Decide if the number itself is allowed. “less than” does not allow the number itself, so there is no “or equal to” line.<
- Write it with the letter on the left. t stands for the temperature in °C.t < 0
Check. Read it back: “t is less than 0” means the same as the sentence. ✓
Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).
Key words. At least means this or more. At most means this or less.
Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.
- Find the key phrase. The phrase tells us which symbol to use.“no more than”
- Decide if the number itself is allowed. “no more than” allows the boundary number itself, so we need the “or equal to” line under the symbol.≤
- Write it with the letter on the left. w stands for the mass of the parcel in kg.w ≤ 2
Check. Read it back: “w is less than or equal to 2” means the same as the sentence. ✓
Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).
Key words. At least means this or more. At most means this or less.
Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.
- Find the key phrase. The phrase tells us which symbol to use.“more than”
- Decide if the number itself is allowed. “more than” does not allow the number itself, so there is no “or equal to” line.>
- Write it with the letter on the left. n stands for the number of people.n > 500
Check. Read it back: “n is greater than 500” means the same as the sentence. ✓
Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).
Key words. At least means this or more. At most means this or less.
Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.
- Find the key phrase. The phrase tells us which symbol to use.“a minimum of”
- Decide if the number itself is allowed. “a minimum of” allows the boundary number itself, so we need the “or equal to” line under the symbol.≥
- Write it with the letter on the left. m stands for the number of marks.m ≥ 40
Check. Read it back: “m is greater than or equal to 40” means the same as the sentence. ✓
Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).
Key words. Greater than means bigger. Less than means smaller.
Method. Say the letter, then the name of the symbol, then the number.
- Name each symbol. < is “less than”, > is “greater than”. A line underneath adds “or equal to”.> means “greater than”
- Say it in a sentence. Read the letter first, then the symbol, then the number.x is greater than 5
Check. Check by drawing a number line: x can be any number bigger than 5. ✓
Watch out. Do not mix up < and >. The open end of the symbol faces the bigger number.
Key words. Greater than means bigger. Less than means smaller.
Method. Say the letter, then the name of the symbol, then the number.
- Name each symbol. < is “less than”, > is “greater than”. A line underneath adds “or equal to”.≤ means “less than or equal to”
- Say it in a sentence. Read the letter first, then the symbol, then the number.n is less than or equal to 3
Check. Check by drawing a number line: n can be any number smaller than 3 (and also 3 itself). ✓
Watch out. Do not mix up < and >. The open end of the symbol faces the bigger number.
Key words. Greater than means bigger. Less than means smaller.
Method. Say the letter, then the name of the symbol, then the number.
- Name each symbol. < is “less than”, > is “greater than”. A line underneath adds “or equal to”.≥ means “greater than or equal to”
- Say it in a sentence. Read the letter first, then the symbol, then the number.y is greater than or equal to 10
Check. Check by drawing a number line: y can be any number bigger than 10 (and also 10 itself). ✓
Watch out. Do not mix up < and >. The open end of the symbol faces the bigger number.
Key words. Greater than means bigger. Less than means smaller.
Method. Say the letter, then the name of the symbol, then the number.
- Name each symbol. < is “less than”, > is “greater than”. A line underneath adds “or equal to”.< means “less than”
- Say it in a sentence. Read the letter first, then the symbol, then the number.t is less than 0.5
Check. Check by drawing a number line: t can be any number smaller than 0.5. ✓
Watch out. Do not mix up < and >. The open end of the symbol faces the bigger number.
Key words. Greater than means bigger. Less than means smaller.
Method. Say the letter, then the name of the symbol, then the number.
- Name each symbol. < is “less than”, > is “greater than”. A line underneath adds “or equal to”.< means “less than”
- Say it in a sentence. Read the letter first, then the symbol, then the number.a is less than −4
Check. Check by drawing a number line: a can be any number smaller than −4. ✓
Watch out. Do not mix up < and >. The open end of the symbol faces the bigger number.
Key words. Greater than means bigger. Less than means smaller.
Method. Say the letter, then the name of the symbol, then the number.
- Name each symbol. < is “less than”, > is “greater than”. A line underneath adds “or equal to”.≥ means “greater than or equal to”
- Say it in a sentence. Read the letter first, then the symbol, then the number.k is greater than or equal to 0
Check. Check by drawing a number line: k can be any number bigger than 0 (and also 0 itself). ✓
Watch out. Do not mix up < and >. The open end of the symbol faces the bigger number.
Key words. A solution is a number that makes the inequality true.
Method. Put each number in place of the letter and decide whether the inequality is true.
- Write the inequality and what it means. We test each number by replacing x with it.x > 3, which says “x is greater than 3”
- Test every number in turn. A number is a solution only if the inequality is true for it. Compare using a number line if you need to.1 > 3? False ✗ not allowed
3 > 3? False ✗ not allowed
3.1 > 3? True ✓ allowed
5 > 3? True ✓ allowed
−4 > 3? False ✗ not allowed - List the numbers that work. Keep only the ones marked true.3.1, 5
Check. The boundary is 3. Numbers to the right of it work, and the boundary itself does not work, which agrees with the list. ✓
Watch out. Check the boundary number 3 carefully. With > it is not a solution.
Key words. A solution is a number that makes the inequality true.
Method. Put each number in place of the letter and decide whether the inequality is true.
- Write the inequality and what it means. We test each number by replacing x with it.x ≤ −2, which says “x is less than or equal to −2”
- Test every number in turn. A number is a solution only if the inequality is true for it. Compare using a number line if you need to.−5 ≤ −2? True ✓ allowed
−2 ≤ −2? True ✓ allowed
−1 ≤ −2? False ✗ not allowed
0 ≤ −2? False ✗ not allowed
−2.5 ≤ −2? True ✓ allowed - List the numbers that work. Keep only the ones marked true.−5, −2, −2.5
Check. The boundary is −2. Numbers to the left of it work, and the boundary itself works, which agrees with the list. ✓
Watch out. Check the boundary number −2 carefully. With ≤ it is a solution.
Key words. At least means this or more. At most means this or less.
Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.
- Find the key phrase. The phrase tells us which symbol to use.“a maximum of”
- Decide if the number itself is allowed. “a maximum of” allows the boundary number itself, so we need the “or equal to” line under the symbol.≤
- Write it with the letter on the left. s stands for the speed in mph.s ≤ 40
Check. Read it back: “s is less than or equal to 40” means the same as the sentence. ✓
Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).
Key words. At least means this or more. At most means this or less.
Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.
- Find the key phrase. The phrase tells us which symbol to use.“at least”
- Decide if the number itself is allowed. “at least” allows the boundary number itself, so we need the “or equal to” line under the symbol.≥
- Write it with the letter on the left. h stands for the height in cm.h ≥ 120
Check. Read it back: “h is greater than or equal to 120” means the same as the sentence. ✓
Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).
Key words. At least means this or more. At most means this or less.
Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.
- Find the key phrase. The phrase tells us which symbol to use.“fewer than”
- Decide if the number itself is allowed. “fewer than” does not allow the number itself, so there is no “or equal to” line.<
- Write it with the letter on the left. c stands for the number of customers.c < 25
Check. Read it back: “c is less than 25” means the same as the sentence. ✓
Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).
Key words. At least means this or more. At most means this or less.
Method. Spot the key phrase, choose the symbol, and decide whether the boundary number is included.
- Find the key phrase. The phrase tells us which symbol to use.“over”
- Decide if the number itself is allowed. “over” does not allow the number itself, so there is no “or equal to” line.>
- Write it with the letter on the left. v stands for the number of volunteers.v > 90
Check. Read it back: “v is greater than 90” means the same as the sentence. ✓
Watch out. “At least” and “at most” include the number (≥ and ≤). “More than” and “less than” do not (> and <).
Key words. A solution is a number that makes the inequality true.
Method. Put each number in place of the letter and decide whether the inequality is true.
- Write the inequality and what it means. We test each number by replacing s with it.s ≤ 40 mph, which says “s is less than or equal to 40”
- Test every number in turn. A number is a solution only if the inequality is true for it. Compare using a number line if you need to.38 mph ≤ 40 mph? True ✓ allowed
40 mph ≤ 40 mph? True ✓ allowed
43 mph ≤ 40 mph? False ✗ not allowed
39.5 mph ≤ 40 mph? True ✓ allowed - List the numbers that work. Keep only the ones marked true.38 mph, 40 mph, 39.5 mph
Check. The boundary is 40. Numbers to the left of it work, and the boundary itself works, which agrees with the list. ✓
Watch out. Check the boundary number 40 carefully. With ≤ it is a solution.
Key words. A solution is a number that makes the inequality true.
Method. Put each number in place of the letter and decide whether the inequality is true.
- Write the inequality and what it means. We test each number by replacing w with it.w ≤ 23 kg, which says “w is less than or equal to 23”
- Test every number in turn. A number is a solution only if the inequality is true for it. Compare using a number line if you need to.22.8 kg ≤ 23 kg? True ✓ allowed
23 kg ≤ 23 kg? True ✓ allowed
23.1 kg ≤ 23 kg? False ✗ not allowed
21.5 kg ≤ 23 kg? True ✓ allowed - List the numbers that work. Keep only the ones marked true.22.8 kg, 23 kg, 21.5 kg
Check. The boundary is 23. Numbers to the left of it work, and the boundary itself works, which agrees with the list. ✓
Watch out. Check the boundary number 23 carefully. With ≤ it is a solution.
Key words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.
Method. Draw the range on a number line, then list only the whole numbers inside it.
- Read the inequality. n is bigger than 2 and smaller than 6. The word integer means a whole number (positive, negative or zero).2 < n < 6
- Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
- Find the smallest and largest integers allowed. The lower end is not included, so go to the next integer above 2. For the upper end: not included, so the integer just below 6.Smallest = 3, largest = 5
- List the integers from smallest to largest. Count up in ones.3, 4, 5
Check. Test the end integers: 3 and 5 satisfy 2 < n < 6, but 2 and 6 do not. ✓
Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.
Key words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.
Method. Draw the range on a number line, then list only the whole numbers inside it.
- Read the inequality. n is bigger than or equal to −3 and smaller than 1. The word integer means a whole number (positive, negative or zero).−3 ≤ n < 1
- Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
- Find the smallest and largest integers allowed. The lower end is included, so start at −3. For the upper end: not included, so the integer just below 1.Smallest = −3, largest = 0
- List the integers from smallest to largest. Count up in ones.−3, −2, −1, 0
Check. Test the end integers: −3 and 0 satisfy −3 ≤ n < 1, but −4 and 1 do not. ✓
Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.
Key words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.
Method. Draw the range on a number line, then list only the whole numbers inside it.
- Read the inequality. n is bigger than 0 and smaller than or equal to 4. The word integer means a whole number (positive, negative or zero).0 < n ≤ 4
- Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
- Find the smallest and largest integers allowed. The lower end is not included, so go to the next integer above 0. For the upper end: included, so up to 4.Smallest = 1, largest = 4
- List the integers from smallest to largest. Count up in ones.1, 2, 3, 4
Check. Test the end integers: 1 and 4 satisfy 0 < n ≤ 4, but 0 and 5 do not. ✓
Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.
Key words. An integer is a whole number, such as 3, 0 or −4. Decimals and fractions are not integers.
Method. Draw the range on a number line, then list only the whole numbers inside it.
- Read the inequality. n is bigger than −5 and smaller than −1. The word integer means a whole number (positive, negative or zero).−5 < n < −1
- Show the possible values on a number line. Mark the integers that lie inside the range. Remember: open circle = not included, closed circle = included.
- Find the smallest and largest integers allowed. The lower end is not included, so go to the next integer above −5. For the upper end: not included, so the integer just below −1.Smallest = −4, largest = −2
- List the integers from smallest to largest. Count up in ones.−4, −3, −2
Check. Test the end integers: −4 and −2 satisfy −5 < n < −1, but −5 and −1 do not. ✓
Watch out. Check each end carefully: < leaves the end number out, ≤ keeps it in.
Method. Think about what each symbol allows.
- Say what ≤ means. ≤ means “less than or equal to”. So the statement is true if the numbers are smaller or the same.5 and 5 are the same, so 5 ≤ 5 is true.
- Say what < means. < means “less than” only. The first number has to be strictly smaller.5 is not smaller than 5, so 5 < 5 is false.
- Write the conclusion. The extra line under ≤ is what lets the numbers be equal.≤ allows “equal to”; < does not.
Check. Test with different numbers: 4 ≤ 5 and 4 < 5 are both true, but only ≤ is true when both numbers are 5. ✓
Watch out. “Or equal to” is the whole point of the line under the symbol.
Method. Compare the two negative numbers using a number line.
- Picture the number line. Numbers get smaller as you move left.−9 is to the left of −3.
- Decide which is bigger. The number further right is bigger, so −3 is bigger than −9.−9 < −3
- Answer. Amir has used the wrong symbol: it should be <.No, he is not correct.
Check. Think of money: owing £9 is worse than owing £3, so −9 is smaller than −3. ✓
Watch out. 9 is bigger than 3, but once the numbers are negative the order reverses.
Method. Work out what “no more than” means, then choose the symbol.
- Think about “no more than 7”. It means x can be 7, or anything smaller than 7. It cannot be bigger than 7.7, 6, 0, −3 are fine; 8 is not.
- Choose the symbol. Values smaller than or equal to 7 need ≤. Ben used ≥, which allows bigger numbers instead.≤
- Write the correct inequality. x ≤ 7
Check. Test x = 10: it is bigger than 7 so it should NOT work. 10 ≤ 7 is false, which is right. ✓
Watch out. “No more than” and “at most” both mean ≤.
Method. Choose boundaries just outside the list, or exactly on the ends, and check which whole numbers fit.
- Look at the smallest and largest numbers. They are 4 and 6. We can put the boundaries on them (using ≤) or just beyond them (using <).smallest 4, largest 6
- Write one possible inequality. Using strict signs, the boundaries must be 3 and 7, because 3 and 7 are not allowed.3 < n < 7
- Check it. List the whole numbers that satisfy 3 < n < 7.4, 5, 6
Check. 4 ≤ n ≤ 6 gives exactly the same whole numbers, 4, 5, 6. ✓
Watch out. There is more than one correct answer. Always list the whole numbers to check.
Method. Try the square of each positive whole number in turn until the square is too big.
- Test n = 1. Work out 12 and compare it with 20.12 = 1, and 1 < 20, so 1 works
- Test n = 2. Work out 22 and compare it with 20.22 = 4, and 4 < 20, so 2 works
- Test n = 3. Work out 32 and compare it with 20.32 = 9, and 9 < 20, so 3 works
- Test n = 4. Work out 42 and compare it with 20.42 = 16, and 16 < 20, so 4 works
- Test n = 5. Work out 52 and compare it with 20.52 = 25, and 25 > 20, so 5 is too big
- Stop and list. The squares keep getting bigger, so no larger number can work.1, 2, 3, 4
Check. The largest value 4 has square 16 < 20, but 5 has square 25 > 20. ✓
Watch out. Positive means bigger than 0, so 0 is not included.