Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Algebra

Equations and Inequalities

Turn word problems into equations, solve equations with brackets, fractions and the unknown on both sides, and use inequalities and number lines to show open and closed intervals.

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

In the curriculum

Write a situation described in words as an equation, write an equation as a situation in words, and solve it. The equations can have whole-number or fraction coefficients, and the unknown can be on one side or on both sides. [8Ae.06]
Understand that a letter can stand for a whole range of numbers (an interval), shown with inequality symbols such as x > 3 or −2 < x ≤ 5 (one or two end values), and show open and closed ends on a number line. [8Ae.07]

An equation is a statement that two things are equal, such as 4x + 7 = 31. Solving it means finding the value of the letter that makes the statement true. In Stage 8 you solve harder equations — with brackets, with fractions and with the letter on both sides — and you turn word problems into equations yourself. You also meet inequalities: a letter that stands for a whole range of numbers, like all the numbers between −2 and 3. Both ideas are used again and again in Stage 9 and in the Checkpoint test, so learn the methods carefully.

1. Key words

Word / phrase	Meaning / use
equation	معادلة — two expressions joined by an = sign, e.g. 3x + 5 = 20
unknown	المجهول — the letter whose value you must find
solve / solution	يحل / الحل — find the value that makes the equation true, e.g. x = 5
inverse operation	العملية العكسية — the operation that undoes another: + undoes −, × undoes ÷
coefficient	المعامل — the number multiplying a letter: in 7x it is 7; in x/4 it is 1/4
expand	يفك الأقواس — multiply out a bracket: 3(x − 4) = 3x − 12
substitute	يعوض — put a number in place of the letter to check
inequality	متباينة — a statement using <, >, ≤ or ≥
interval	فترة — all the numbers between two end values
open / closed end	طرف مفتوح / مغلق — the end value is NOT included (○) / IS included (●)
integer	عدد صحيح — a whole number: … −2, −1, 0, 1, 2 …

2. Expression, formula or equation?

Letters are used in three different ways, and it helps to know which one you are looking at.

Type	Example	What the letter means
Expression	4x + 7	No = sign. x can be any number; you can simplify or evaluate it, but not solve it.
Formula	P = 2l + 2w	A rule connecting quantities. Each letter stands for a real quantity (perimeter, length, width).
Equation	4x + 7 = 31	True only for a particular value of x (here x = 6). You solve it.

3. The balance method

Think of an equation as a pair of balanced scales. The two sides weigh the same. If you add, subtract, multiply or divide on one side, you must do exactly the same to the other side, or the scales stop balancing. Your aim is to get the letter on its own on one side.

The equation 2x + 3 = 11 as balanced scales. Each green box is x and each yellow circle is 1.

Method for solving any linear equation

Simplify each side: expand any brackets and collect like terms.
Letter on both sides? Subtract the smaller letter term from both sides, so the letter is on one side only.
Undo + or − by doing the inverse to both sides.
Undo × or ÷ by doing the inverse to both sides.
Check: substitute your answer into the ORIGINAL equation. Both sides must give the same number.

Example: 2x + 3 = 11 → subtract 3 from both sides: 2x = 8 → divide both sides by 2: x = 4. Check: 2 × 4 + 3 = 11 ✓. Write one step per line with the = signs underneath each other — it is clearer, and examiners can give you method marks even if you slip later.

4. Brackets, negative numbers and fraction answers

Equation	Working	Solution
3(x − 4) = 15	Expand: 3x − 12 = 15 → add 12: 3x = 27 → ÷ 3	x = 9
3(x − 4) = 15 (another way)	Divide both sides by 3 first: x − 4 = 5 → add 4	x = 9
20 − 3x = 8	Add 3x to both sides: 20 = 8 + 3x → subtract 8: 12 = 3x → ÷ 3	x = 4
4x + 9 = 1	Subtract 9: 4x = −8 → ÷ 4	x = −2
6x = 15	÷ 6: x = 15/6 = 5/2	x = 5/2 = 2 1/2 = 2.5

Answers do not have to be positive whole numbers. A solution can be negative (x = −2) or a fraction (x = 5/2). Give a fraction answer in its simplest form, or as an exact decimal. Do not round 1/3 to 0.3 — that is not exact.

5. Equations with fractional coefficients

A fractional coefficient means the letter is multiplied by a fraction, for example (2/3)x, or divided by a number, for example x/5 (which is the same as (1/5)x). To clear the fraction, multiply both sides by the denominator.

Equation	Working	Solution
x/5 = 3	Multiply both sides by 5	x = 15
x/4 + 6 = 9	Subtract 6: x/4 = 3 → × 4	x = 12
(2/3)x = 10	× 3: 2x = 30 → ÷ 2 (or multiply by 3/2 in one step)	x = 15
(3/4)x − 2 = 10	Add 2: (3/4)x = 12 → × 4: 3x = 48 → ÷ 3	x = 16
(x + 7)/3 = 4	× 3: x + 7 = 12 → subtract 7	x = 5
(3x − 1)/4 = 5	× 4: 3x − 1 = 20 → add 1: 3x = 21 → ÷ 3	x = 7

The fraction line is a bracket

In (x + 7)/3 the whole of x + 7 is divided by 3. So you cannot subtract the 7 first. Multiply both sides by 3 first, which removes the fraction and leaves x + 7 = 12.

6. The unknown on both sides

When the letter appears on both sides, first remove the letter from one side. A good habit is to subtract the smaller letter term, so the letter term that is left is positive.

Equation	Working	Check
5x + 3 = 2x + 18	Subtract 2x: 3x + 3 = 18 → subtract 3: 3x = 15 → x = 5	Left 5×5+3 = 28; right 2×5+18 = 28 ✓
7 − 2x = 3x − 8	Add 2x (−2x is smaller than 3x): 7 = 5x − 8 → add 8: 15 = 5x → x = 3	Left 7−6 = 1; right 9−8 = 1 ✓
4(x + 2) = 6x − 5	Expand: 4x + 8 = 6x − 5 → subtract 4x: 8 = 2x − 5 → 13 = 2x → x = 6.5	Left 4×8.5 = 34; right 39−5 = 34 ✓
(1/2)x + 5 = 2x − 1	Multiply every term by 2: x + 10 = 4x − 2 → subtract x: 10 = 3x − 2 → 12 = 3x → x = 4	Left 2+5 = 7; right 8−1 = 7 ✓

Notice that it does not matter if the letter ends up on the right: 15 = 5x is just as good as 5x = 15. The answer is still x = 3.

7. From words to equations — and back again

Many Checkpoint questions describe a situation in words. You must write the equation yourself.

Choose a letter for the unknown and say what it stands for (e.g. "Let p be the price of one pen in dollars").
Write each quantity in the problem as an expression in that letter.
Find the two things that are equal and join them with = to make the equation.
Solve the equation.
Answer the question that was asked, in words, with units — and check it makes sense in the story.

Words	Algebra
5 more than n	n + 5
5 less than n	n − 5 (not 5 − n)
n subtracted from 5	5 − n
twice n / double n	2n
a third of n	n/3 or (1/3)n
the sum of n and 4, then multiplied by 3	3(n + 4)
I think of a number, multiply it by 4 and subtract 7. The answer is 25.	4n − 7 = 25
Three times a number is equal to the number plus 18.	3n = n + 18

Going the other way. You may be asked to write a story for an equation. For 2x + 15 = 60 you could write: "A book costs x dollars. Two books and a bag costing $15 cost $60 altogether." Any story is fine as long as every part of the equation is explained: 2x is two books, + 15 is the bag, = 60 is the total.

8. Inequalities and intervals

Sometimes a letter stands for many values, not just one. "The temperature was above 2 °C" does not give one number — it gives a whole range. We write this with an inequality: t > 2.

Symbol	Meaning	In words you may see	End value on a number line
x < 4	x is less than 4	below, under, fewer than	open circle ○ — 4 is not included
x > 4	x is greater than 4	more than, over, above	open circle ○ — 4 is not included
x ≤ 4	x is less than or equal to 4	at most, no more than, maximum	filled circle ● — 4 is included
x ≥ 4	x is greater than or equal to 4	at least, no less than, minimum	filled circle ● — 4 is included

Four intervals. An open circle (○) means that end value is not included; a filled circle (●) means it is included. An arrow means the interval goes on for ever.

A two-term inequality such as −2 < x ≤ 3 gives both ends of the interval. Read it from the middle outwards: "x is greater than −2 and less than or equal to 3". The smaller number is always on the left. On the number line it is a line segment between −2 and 3, with an open circle at −2 and a filled circle at 3.

Closed interval — both ends included, e.g. −3 ≤ x ≤ 1 (two filled circles).
Open interval — neither end included, e.g. −1 < x < 4 (two open circles).
One end open and one closed, e.g. −2 < x ≤ 3.
x can be any number in the interval — 2.7, 1/2, −1.99 — not only whole numbers. If the question says "x is an integer", list only the whole numbers: for −2 < x ≤ 3 they are −1, 0, 1, 2, 3 (five integers; −2 is not included).
Real life: "Children aged at least 8 and under 14 can join the club" → 8 ≤ a < 14.

افتكر القاعدة دي

المعادلة زي الميزان: أي حاجة تعملها في ناحية لازم تعملها في الناحية التانية بالظبط.
استخدم العملية العكسية: الطرح عكس الجمع، والقسمة عكس الضرب. ولو المجهول في الناحيتين، اطرح الحد الأصغر من الناحيتين الأول.
لو فيه كسر، اضرب الطرفين في المقام عشان الكسر يختفي.

9. Worked examples

Worked example 1 — Solve 5(x − 3) = 2x + 9. Show your working. [3]

Step	Working	What I did
1	5x − 15 = 2x + 9	Expanded the bracket: 5 × x and 5 × (−3)
2	3x − 15 = 9	Subtracted 2x from both sides
3	3x = 24	Added 15 to both sides
4	x = 8	Divided both sides by 3
Check	5(8 − 3) = 25 and 2 × 8 + 9 = 25 ✓	Substituted into the original equation

Answer: x = 8. Common mistake: expanding 5(x − 3) as 5x − 3. The 5 multiplies both terms in the bracket.

Worked example 2 — Solve (2x + 5)/3 = 9. [2]

Multiply both sides by 3: 2x + 5 = 27.
Subtract 5: 2x = 22.
Divide by 2: x = 11.
Check: (22 + 5)/3 = 27/3 = 9 ✓

Common mistake: writing 2x/3 + 5 = 9. The 5 is also divided by 3, so clear the fraction first.

Worked example 3 — Solve (1/3)x + 4 = x − 6. [3]

Multiply every term on both sides by 3: x + 12 = 3x − 18.
Subtract x from both sides: 12 = 2x − 18.
Add 18: 30 = 2x.
Divide by 2: x = 15.
Check: left (1/3) × 15 + 4 = 5 + 4 = 9; right 15 − 6 = 9 ✓

Common mistake: multiplying only the fraction term by 3 and forgetting the other terms. Every term must be multiplied.

Worked example 4 — The length of a rectangle is (2x + 3) cm and its width is (x − 1) cm. The perimeter is 40 cm. (a) Write an equation in x. [1] (b) Solve it. [2] (c) Find the length of the rectangle. [1]

The rectangle in Worked example 4.

Part	Working
(a)	Perimeter = 2 × length + 2 × width, so 2(2x + 3) + 2(x − 1) = 40
(b)	4x + 6 + 2x − 2 = 40 → 6x + 4 = 40 → 6x = 36 → x = 6
(c)	Length = 2 × 6 + 3 = 15 cm. (Width = 5 cm; check: 2 × 15 + 2 × 5 = 40 ✓)

Common mistake: stopping at x = 6. The question asks for the length, so substitute back.

Worked example 5 — Omar is three times as old as his sister Mona. In 8 years' time, Omar will be twice as old as Mona. How old is Mona now? [3]

Step	Working
Choose a letter	Let Mona's age now be m years. Then Omar is 3m.
In 8 years	Mona: m + 8. Omar: 3m + 8.
Equation	Omar = twice Mona: 3m + 8 = 2(m + 8)
Solve	3m + 8 = 2m + 16 → m + 8 = 16 → m = 8
Answer and check	Mona is 8 years old (Omar is 24). In 8 years: 32 and 16, and 32 = 2 × 16 ✓

Worked example 6 — (a) Write the inequality shown on the number line. [1] (b) x is an integer. List all the possible values of x. [1] (c) Is 4.5 in this interval? Explain. [1]

Number line for Worked example 6.

Part	Working and answer
(a)	Open circle at −1 (not included) and filled circle at 4 (included): −1 < x ≤ 4
(b)	0, 1, 2, 3, 4 (not −1, because −1 is not included)
(c)	No — the largest value allowed is 4, and 4.5 > 4.

Common mistakes

Doing something to one side only. Every step must be done to both sides (and to every term on a side).
Expanding brackets wrongly: 3(x − 4) is 3x − 12, not 3x − 4. Watch the signs: −2(x − 5) = −2x + 10.
Sign slips when removing a letter term: from 5x = 2x + 18, subtracting 2x gives 3x = 18 (not 7x).
With (2x + 5)/3, subtracting the 5 before multiplying by 3. Clear the fraction first.
Writing "5 less than n" as 5 − n. It is n − 5.
Leaving answers like 12/3 (simplify to 4) or rounding 1/3 to 0.3 (give the exact fraction).
Mixing up open and closed circles, or writing the smaller number on the right (3 < x < −2 is wrong; write −2 < x < 3).
Including an end value that is not allowed when listing integers, e.g. listing −2 for −2 < x ≤ 3.
Finding x but not answering the question (e.g. the question asked for the length or the age of the other person).

How Checkpoint asks this

"Solve …" with [2] or [3] marks: show every line. Marks are usually given for correct steps, such as a correct expansion or a correct rearrangement, as well as the answer.
"Write an equation and solve it to find …": define your letter, write the equation, solve, then answer the question in words.
"Ahmed says the solution is x = 4. Is he correct? Explain." Substitute 4 into both sides and show they are not equal (or solve and compare).
"Draw a ring around the inequality shown on the number line" or "Show −1 ≤ x < 3 on the number line": check each end — open or filled circle.
"x is an integer. Write down all the possible values of x": look carefully at < and ≤ at each end.
Always check your solution by substituting — it takes 20 seconds and catches most mistakes.

المتباينات بالعربي

الدائرة الفاضية ○ يعني الرقم ده مش داخل (< أو >). الدائرة المليانة ● يعني الرقم داخل (≤ أو ≥).
−2 < x ≤ 3 معناها: x أكبر من −2 وأصغر من أو يساوي 3. الرقم الصغير دايمًا على الشمال.
"at least" يعني ≥ (على الأقل)، و"at most" يعني ≤ (على الأكثر).

Practice

Remember

Balance method: whatever you do to one side, do to the other — and to every term.
Order: expand brackets → letters on one side (remove the smaller letter term) → undo + / − → undo × / ÷ → check.
Fractions: multiply both sides (every term) by the denominator to clear them.
Word problems: define the letter, write the equation, solve, answer the question in words with units.
Inequalities: ○ = not included (< >), ● = included (≤ ≥); smaller number on the left; read "at least" as ≥ and "at most" as ≤.

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