GCSE Maths: Forming and solving inequalities from context (Foundation)
Part of the GCSE Maths inequalities chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Forming and solving inequalities from context
Many exam questions describe a real situation in words. You must turn the words into an inequality, solve it, and then explain what your answer means in the situation. Follow these steps.
- Choose a letter for the unknown and say what it stands for.
- Write an expression for each amount in the problem (for example, 7 tickets at £n each costs 7n pounds).
- Choose the sign from the key phrase: “at most” is ≤, “at least” is ≥, “less than” is <, “more than” is >.
- Solve the inequality.
- Interpret the answer: if you are counting things, the answer must be a whole number, and you must round the correct way. Then check it in the original problem.
Rounding the correct way. If the solution is n ≤ 6.29 (tickets), you cannot buy 6.29 tickets. The largest whole number that works is 6. If the solution is n ≥ 18.67 (weeks), the smallest whole number that works is 19, because 18 weeks is not enough. Always round so that the condition is still true, not to the nearest number.
Totals and means. A total is built by adding the pieces. A mean is the total divided by the number of values, so “the mean of four marks is at least 65” becomes (total of marks) ÷ 4 ≥ 65.
Key ideas
- Choose a letter and say what it means.
- Key phrases: at most = ≤, at least = ≥, less than = <, more than = >.
- Solve, then interpret in context.
- For counting problems, the answer must be a whole number: round down for an upper limit and up for a lower limit.
- Always check the answer by substituting back into the story.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let n be the number of tickets Sam buys.
- Write an inequality. The tickets cost 7n pounds, and the drink costs £6. The total must be at most £50, so we use ≤.7n + 6 ≤ 50
- Get the plain numbers on the other side. Subtract 6 from both sides. This is the inverse of adding 6.7n ≤ 44
- Divide both sides by 7. 7n means 7 lots of n. The inverse of multiplying by 7 is dividing by 7. 7 is positive, so the inequality sign stays the same.n ≤ 44 ÷ 7 = 447
- Answer the question in words. The inequality says the number is less than or equal to 447. Because the number of tickets must be a whole number, we need the largest whole number that fits.6 tickets
Check. Test a value inside the solution: n = 6 gives left side = 7 × 6 + 6 = 48, right side = 50. Is 48 ≤ 50? Yes, it is true. ✓
Test a value outside it: n = 7 gives left side = 7 × 7 + 6 = 55, right side = 50. Is 55 ≤ 50? No, it is false, so 7 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let x cm be the unknown length.
- Write an expression for the perimeter. A rectangle has two lengths and two widths, so the perimeter is 2(x + 4) + 2 × 3. Expand: 2x + 8 + 6.Perimeter = 2x + 14
- Get the plain numbers on the other side. Subtract 14 from both sides. This is the inverse of adding 14.2x < 16
- Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.x < 16 ÷ 2 = 8
- Answer the question in words. The length is (x + 4) cm, so x is just a number. It does not have to be whole.x < 8 cm
Check. Test a value inside the solution: x = 7 gives left side = 2 × 7 + 14 = 28, right side = 30. Is 28 < 30? Yes, it is true. ✓
Test a value outside it: x = 9 gives left side = 2 × 9 + 14 = 32, right side = 30. Is 32 < 30? No, it is false, so 9 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let x be Tom’s mark in the fourth test.
- Write an inequality. The total of the four marks is 62 + 70 + 58 + x = (190 + x). The mean is the total divided by 4, and it must be at least 65.(x + 190) ÷ 4 ≥ 65
- Multiply both sides by 4 to get rid of the fraction. Multiplying by the denominator cancels the division. 4 is a positive number, so the inequality sign does not change.x + 190 ≥ 260
- Get the plain numbers on the other side. Subtract 190 from both sides. This is the inverse of adding 190.x ≥ 70
- Answer the question in words. Tom needs at least this mark in the fourth test.x ≥ 70 marks
Check. Test a value inside the solution: x = 71 gives left side = 71 + 1904 = 65.25, right side = 65. Is 65.25 ≥ 65? Yes, it is true. ✓
Test a value outside it: x = 69 gives left side = 69 + 1904 = 64.75, right side = 65. Is 64.75 ≥ 65? No, it is false, so 69 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let h be the number of hours.
- Write an expression for each plan. Plan A costs 10 + 4h pounds. Plan B costs 25 + h pounds. We want A to be cheaper, so A < B.4h + 10 < h + 25
- Get the h terms on one side. There are h terms on both sides. We remove the smaller one, so that the h term left over is positive and we do not need to divide by a negative number later.Subtract h from both sides:
3h + 10 < 25 - Get the plain numbers on the other side. Subtract 10 from both sides. This is the inverse of adding 10.3h < 15
- Divide both sides by 3. 3h means 3 lots of h. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.h < 15 ÷ 3 = 5
- Answer the question in words. Plan A is cheaper for fewer than this number of hours.h < 5 hours
Check. Test a value inside the solution: h = 4 gives left side = 4 × 4 + 10 = 26, right side = 4 + 25 = 29. Is 26 < 29? Yes, it is true. ✓
Test a value outside it: h = 6 gives left side = 4 × 6 + 10 = 34, right side = 6 + 25 = 31. Is 34 < 31? No, it is false, so 6 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Watch out
The answer must make sense. If the answer is a number of people, tickets or cakes it must be a whole number, and you must round the right way: n ≤ 6.29 means at most 6, not 7. Also, do not use the nearest whole number blindly: n ≥ 18.67 means at least 19.Exam tip
Write down what your letter stands for in a short sentence at the start. Examiners award marks for defining the variable, and it helps you avoid mixing up quantities.Exercise 10G Inequalities in context
Non-calculatorKey words. To form an inequality means to write it from the words, without solving it yet.
Method. Define the letter, write an expression for each part, then choose the sign from the key phrase.
- Write the cost of the pens. Each pen costs £3, so n pens cost 3n.3n
- Add the book. The total cost is the pens plus the book.3n + 5
- Choose the sign. He has £40, so the cost can be at most 40.3n + 5 ≤ 40
Check. Substitute a sensible value into your inequality to see if it matches the story. The expression for the unknown total is the left side, and the limit is the right side. ✓
Watch out. Look for the key phrase (at most, at least, less than, more than) to choose the sign.
Key words. To form an inequality means to write it from the words, without solving it yet.
Method. Define the letter, write an expression for each part, then choose the sign from the key phrase.
- Write the cost of the miles. £2 for each mile, so m miles cost 2m.2m
- Add the fixed charge. The £4 is charged once.2m + 4
- Choose the sign. “Less than 30” has no equals line.2m + 4 < 30
Check. Substitute a sensible value into your inequality to see if it matches the story. The expression for the unknown total is the left side, and the limit is the right side. ✓
Watch out. Look for the key phrase (at most, at least, less than, more than) to choose the sign.
Key words. To form an inequality means to write it from the words, without solving it yet.
Method. Define the letter, write an expression for each part, then choose the sign from the key phrase.
- Multiply by 5. Five times the number.5x
- Subtract 8. 5x − 8
- Choose the sign. “At least 22” means 22 or more.5x − 8 ≥ 22
Check. Substitute a sensible value into your inequality to see if it matches the story. The expression for the unknown total is the left side, and the limit is the right side. ✓
Watch out. Look for the key phrase (at most, at least, less than, more than) to choose the sign.
Key words. To form an inequality means to write it from the words, without solving it yet.
Method. Define the letter, write an expression for each part, then choose the sign from the key phrase.
- Write the perimeter. A square has four equal sides.4s
- Choose the sign. “Greater than 50” has no equals line.4s > 50
Check. Substitute a sensible value into your inequality to see if it matches the story. The expression for the unknown total is the left side, and the limit is the right side. ✓
Watch out. Look for the key phrase (at most, at least, less than, more than) to choose the sign.
Key words. To form an inequality means to write it from the words, without solving it yet.
Method. Define the letter, write an expression for each part, then choose the sign from the key phrase.
- Write the perimeter. Two lengths and two widths.2(x + 3) + 2 × 5
- Simplify. 2x + 6 + 102x + 16
- Choose the sign. “At most 36” includes 36.2x + 16 ≤ 36
Check. Substitute a sensible value into your inequality to see if it matches the story. The expression for the unknown total is the left side, and the limit is the right side. ✓
Watch out. Look for the key phrase (at most, at least, less than, more than) to choose the sign.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let n be the number of pens.
- Write an inequality. 3n pounds for the pens, £5 for the book, at most £40.3n + 5 ≤ 40
- Get the plain numbers on the other side. Subtract 5 from both sides. This is the inverse of adding 5.3n ≤ 35
- Divide both sides by 3. 3n means 3 lots of n. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.n ≤ 35 ÷ 3 = 353
- Answer the question in words. The inequality says the number is less than or equal to 353. Because a person cannot buy part of a pen, we need the largest whole number that fits.11 pens
Check. Test a value inside the solution: n = 11 gives left side = 3 × 11 + 5 = 38, right side = 40. Is 38 ≤ 40? Yes, it is true. ✓
Test a value outside it: n = 12 gives left side = 3 × 12 + 5 = 41, right side = 40. Is 41 ≤ 40? No, it is false, so 12 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let m be the length of the journey in miles.
- Write an inequality. Cost = 4 + 2m, and this must be less than 30.2m + 4 < 30
- Get the plain numbers on the other side. Subtract 4 from both sides. This is the inverse of adding 4.2m < 26
- Divide both sides by 2. 2m means 2 lots of m. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.m < 26 ÷ 2 = 13
- Answer the question in words. A journey can be any number of miles, so we leave the answer as an inequality.m < 13 miles
Check. Test a value inside the solution: m = 12 gives left side = 2 × 12 + 4 = 28, right side = 30. Is 28 < 30? Yes, it is true. ✓
Test a value outside it: m = 14 gives left side = 2 × 14 + 4 = 32, right side = 30. Is 32 < 30? No, it is false, so 14 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let n be the number of boxes in the lift.
- Write an inequality. Each box weighs 12 kg, so the boxes weigh 12n kg. Sam weighs 80 kg. The total must be at most 200 kg.12n + 80 ≤ 200
- Get the plain numbers on the other side. Subtract 80 from both sides. This is the inverse of adding 80.12n ≤ 120
- Divide both sides by 12. 12n means 12 lots of n. The inverse of multiplying by 12 is dividing by 12. 12 is positive, so the inequality sign stays the same.n ≤ 120 ÷ 12 = 10
- Answer the question in words. The inequality says the number is less than or equal to 10. Because the number of boxes must be a whole number, we need the largest whole number that fits.10 boxes
Check. Test a value inside the solution: n = 9 gives left side = 12 × 9 + 80 = 188, right side = 200. Is 188 ≤ 200? Yes, it is true. ✓
Test a value outside it: n = 11 gives left side = 12 × 11 + 80 = 212, right side = 200. Is 212 ≤ 200? No, it is false, so 11 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let n be the number of cakes sold.
- Write an inequality. Profit = money from sales − cost = 3n − 25. The profit must be at least £50.3n − 25 ≥ 50
- Get the plain numbers on the other side. Add 25 to both sides. This is the inverse of subtracting 25.3n ≥ 75
- Divide both sides by 3. 3n means 3 lots of n. The inverse of multiplying by 3 is dividing by 3. 3 is positive, so the inequality sign stays the same.n ≥ 75 ÷ 3 = 25
- Answer the question in words. The inequality says the number is greater than or equal to 25. Because the number of cakes must be a whole number, we need the smallest whole number that fits.25 cakes
Check. Test a value inside the solution: n = 26 gives left side = 3 × 26 − 25 = 53, right side = 50. Is 53 ≥ 50? Yes, it is true. ✓
Test a value outside it: n = 24 gives left side = 3 × 24 − 25 = 47, right side = 50. Is 47 ≥ 50? No, it is false, so 24 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let w be the number of weeks.
- Write an inequality. After w weeks she has 120 + 15w pounds, and she wants at least £400.15w + 120 ≥ 400
- Get the plain numbers on the other side. Subtract 120 from both sides. This is the inverse of adding 120.15w ≥ 280
- Divide both sides by 15. 15w means 15 lots of w. The inverse of multiplying by 15 is dividing by 15. 15 is positive, so the inequality sign stays the same.w ≥ 280 ÷ 15 = 563
- Answer the question in words. The inequality says the number is greater than or equal to 563. Because she needs whole weeks and 18 weeks would not be enough, we need the smallest whole number that fits.19 weeks
Check. Test a value inside the solution: w = 19 gives left side = 15 × 19 + 120 = 405, right side = 400. Is 405 ≥ 400? Yes, it is true. ✓
Test a value outside it: w = 18 gives left side = 15 × 18 + 120 = 390, right side = 400. Is 390 ≥ 400? No, it is false, so 18 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let x be Ali’s age in years. His sister is 3 years older, so her age is x + 3.
- Write an inequality. The sum of their ages is x + (x + 3) = 2x + 3. It is less than 30.2x + 3 < 30
- Get the plain numbers on the other side. Subtract 3 from both sides. This is the inverse of adding 3.2x < 27
- Divide both sides by 2. 2x means 2 lots of x. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.x < 27 ÷ 2 = 13.5
- Answer the question in words. The inequality says the number is less than 13.5. Because Ali’s age is given as a whole number of years, we need the largest whole number that fits.13 years old
Check. Test a value inside the solution: x = 13 gives left side = 2 × 13 + 3 = 29, right side = 30. Is 29 < 30? Yes, it is true. ✓
Test a value outside it: x = 14 gives left side = 2 × 14 + 3 = 31, right side = 30. Is 31 < 30? No, it is false, so 14 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let m be the number of months.
- Write an expression for each gym. Gym A: 15 + 8m. Gym B: 5 + 12m. We want A to be cheaper.8m + 15 < 12m + 5
- Get the m terms on one side. There are m terms on both sides. We remove the smaller one, so that the m term left over is positive and we do not need to divide by a negative number later.Subtract 8m from both sides:
15 < 4m + 5 - Get the plain numbers on the other side. Subtract 5 from both sides.10 < 4m
- Divide both sides by 4. The inverse of multiplying by 4 is dividing by 4. 4 is positive, so the inequality sign stays the same.10 ÷ 4 = 2.5
2.5 < m - Write it with m on the left. 2.5 < m says the same thing as m > 2.5: the sign still points to the smaller number.m > 2.5
- Answer the question in words. The inequality says the number is greater than 2.5. Because we need a whole number of months, we need the smallest whole number that fits.3 months
Check. Test a value inside the solution: m = 3 gives left side = 8 × 3 + 15 = 39, right side = 12 × 3 + 5 = 41. Is 39 < 41? Yes, it is true. ✓
Test a value outside it: m = 2 gives left side = 8 × 2 + 15 = 31, right side = 12 × 2 + 5 = 29. Is 31 < 29? No, it is false, so 2 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let x be the mark in the fourth test.
- Write an inequality. Total = 64 + 71 + 58 + x = x + 193. The mean is the total divided by 4 and it must be at least 65.(x + 193) ÷ 4 ≥ 65
- Multiply both sides by 4 to get rid of the fraction. Multiplying by the denominator cancels the division. 4 is a positive number, so the inequality sign does not change.x + 193 ≥ 260
- Get the plain numbers on the other side. Subtract 193 from both sides. This is the inverse of adding 193.x ≥ 67
- Answer the question in words. This is the least mark that gives a mean of 65 or more.x ≥ 67 marks
Check. Test a value inside the solution: x = 68 gives left side = 68 + 1934 = 65.25, right side = 65. Is 65.25 ≥ 65? Yes, it is true. ✓
Test a value outside it: x = 66 gives left side = 66 + 1934 = 64.75, right side = 65. Is 64.75 ≥ 65? No, it is false, so 66 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let n be the number.
- Write an inequality. Double the number is 2n. Subtract 7. The result is greater than 15.2n − 7 > 15
- Get the plain numbers on the other side. Add 7 to both sides. This is the inverse of subtracting 7.2n > 22
- Divide both sides by 2. 2n means 2 lots of n. The inverse of multiplying by 2 is dividing by 2. 2 is positive, so the inequality sign stays the same.n > 22 ÷ 2 = 11
- Answer the question in words. The inequality says the number is greater than 11. Because the question says the number is a whole number, we need the smallest whole number that fits.12
Check. Test a value inside the solution: n = 12 gives left side = 2 × 12 − 7 = 17, right side = 15. Is 17 > 15? Yes, it is true. ✓
Test a value outside it: n = 10 gives left side = 2 × 10 − 7 = 13, right side = 15. Is 13 > 15? No, it is false, so 10 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Key words. An inequality compares two quantities using < (less than), > (greater than), ≤ (less than or equal to) or ≥ (greater than or equal to). To solve an inequality means to find all the values of the letter that make it true. The answer is usually a whole range of values, not just one number.
Method. Turn the words into an inequality, solve it, then say what the answer means in the context of the question.
- Choose a letter. Let x be a length in cm.
- Write an inequality. The perimeter is (2x + 1) + (2x + 1) + 9 = 4x + 11. It is less than 40.4x + 11 < 40
- Get the plain numbers on the other side. Subtract 11 from both sides. This is the inverse of adding 11.4x < 29
- Divide both sides by 4. 4x means 4 lots of x. The inverse of multiplying by 4 is dividing by 4. 4 is positive, so the inequality sign stays the same.x < 29 ÷ 4 = 7.25
- Answer the question in words. The side lengths are in cm and do not have to be whole numbers.x < 7.25
Check. Test a value inside the solution: x = 7 gives left side = 4 × 7 + 11 = 39, right side = 40. Is 39 < 40? Yes, it is true. ✓
Test a value outside it: x = 8 gives left side = 4 × 8 + 11 = 43, right side = 40. Is 43 < 40? No, it is false, so 8 is correctly left out.
Watch out. Always finish by answering the question that was asked, using the right units, and check whether the answer must be a whole number.
Method. Think about what n stands for.
- What does n count? It counts tickets, and you cannot buy part of a ticket.n must be a whole number
- Which whole numbers satisfy n ≤ 6.29? 6 does (6 ≤ 6.29). 7 does not (7 > 6.29).Greatest whole number = 6
- Write the answer in words. 6 tickets
Check. 6 tickets cost 6 × 7 + 6 = 48 ≤ 50. 7 tickets would cost 7 × 7 + 6 = 55 > 50. ✓
Watch out. For an upper limit round down; for a lower limit round up.
Method. Decide what must be true about the total cost.
- Think about the money. The student has £15. The total cost cannot be more than the money available.cost ≤ 15
- Write the cost. n notebooks at £2 each cost 2n.2n
- Write the correct inequality. The cost is at most £15.2n ≤ 15
Check. With n = 8, cost = £16, which is more than £15. 2n ≤ 15 correctly rules this out (16 ≤ 15 is false). ✓
Watch out. “I have £15” sets an upper limit on the spending.
Method. Write an inequality for each pair of sides, solve them, and keep the values that satisfy all of them.
- Use the 5 and 8 sides together. Their sum must be greater than the third side.5 + 8 > x, so 13 > x, which is x < 13
- Use the 5 and x sides together. Their sum must be greater than the 8 side.5 + x > 8, so x > 3, which is x > 3
- Use the 8 and x sides together. Their sum must be greater than 5.8 + x > 5, so x > −3. This is already true if x > 3, so it adds nothing new.
- Combine. Keep values satisfying both x < 13 and x > 3.3 < x < 13
Check. Try x = 10: sides 5, 8, 10 make a triangle (5 + 8 = 13 > 10, 5 + 10 > 8, 8 + 10 > 5). Try x = 2: 2 + 5 = 7, which is less than 8, so it fails. ✓
Watch out. Write the triangle rule for every pair of sides.