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Solving Linear Equations and Inequalities

Solving equations with brackets, fractions and unknowns on both sides, forming equations, and solving inequalities.

  • 10 key terms
  • All boards

Learning Objectives

  1. 1Solve linear equations using inverse operations, including equations with brackets and unknowns on both sides.
  2. 2Solve equations that contain fractions.
  3. 3Form equations from worded and geometric problems and solve them.
  4. 4Solve linear inequalities, show solutions on a number line and list integer solutions.

An equation is a balance

An equation says that two sides are equal, so whatever you do to one side you must do to the other to keep it balanced. Solving means finding the value of the unknown that makes the statement true. Written working matters here: examiners award method marks for each correct step, so a line-by-line layout with the same operation shown on both sides is worth the extra seconds.

Solving by inverse operations

Undo the operations in reverse order, starting with the addition or subtraction.

  • One step

    \(x + 7 = 12\) means subtract 7 from both sides, so \(x = 5\). \(x - 4 = 9\) means add 4, so \(x = 13\).

  • Two steps

    \(3x - 4 = 11\). Add 4 to get \(3x = 15\), then divide by 3 to get \(x = 5\).

  • Fractions

    \(\dfrac{x}{4} + 3 = 8\). Subtract 3 to get \(\dfrac{x}{4} = 5\), then multiply by 4 to get \(x = 20\).

  • A negative coefficient

    \(5 - 2x = 11\). Subtract 5 to get \(-2x = 6\), then divide by \(-2\) to get \(x = -3\).

Unknowns on both sides

Solve \(5x + 3 = 2x + 18\).

Show the solutionHide the solution
  1. 1 Collect the \(x\) terms on one side Subtract \(2x\) from both sides: \(3x + 3 = 18\).
  2. 2 Remove the number from that side Subtract 3 from both sides: \(3x = 15\).
  3. 3 Divide \(x = 5\).
  4. 4 Check Left side: \(5 \times 5 + 3 = 28\). Right side: \(2 \times 5 + 18 = 28\). The sides match.

Answer\(x = 5\)

Equations with brackets

Solve \(3(2x - 1) = 4x + 9\).

Show the solutionHide the solution
  1. 1 Expand the bracket \(6x - 3 = 4x + 9\).
  2. 2 Collect the \(x\) terms Subtract \(4x\) from both sides: \(2x - 3 = 9\).
  3. 3 Add 3 \(2x = 12\).
  4. 4 Divide \(x = 6\). Check: \(3(2 \times 6 - 1) = 33\) and \(4 \times 6 + 9 = 33\).

Answer\(x = 6\)

Equations with fractions on both sides

Solve \(\dfrac{x + 5}{3} = \dfrac{x - 1}{2}\).

Show the solutionHide the solution
  1. 1 Multiply both sides by the denominators Multiply by 6, the common multiple of 3 and 2: \(2(x + 5) = 3(x - 1)\).
  2. 2 Expand the brackets \(2x + 10 = 3x - 3\).
  3. 3 Collect the \(x\) terms Subtract \(2x\) from both sides: \(10 = x - 3\).
  4. 4 Solve Add 3 to both sides: \(x = 13\). Check: \(\dfrac{18}{3} = 6\) and \(\dfrac{12}{2} = 6\).

Answer\(x = 13\)

Forming equations

Many exam questions describe a situation, and you must turn it into an equation first.

  • Choose a letter

    Say what it stands for, such as "let the width be \(x\) cm".

  • Write each quantity

    Express every other quantity in terms of \(x\), such as a length of \((3x + 2)\) cm.

  • Use the fact you are given

    A perimeter, a total, the angle sum of a triangle (180°) or a statement that two things are equal gives you the equation.

  • Answer the question asked

    Solving gives \(x\), but the question may ask for an area or an age, so reread it.

Forming and solving an equation

A rectangle has length \((3x + 2)\) cm and width \((x + 4)\) cm. Its perimeter is 52 cm. Work out the value of \(x\) and the length of the rectangle.

Show the solutionHide the solution
  1. 1 Write the perimeter Perimeter \(= 2 \times (\text{length} + \text{width}) = 2(3x + 2 + x + 4) = 2(4x + 6)\).
  2. 2 Make an equation \(2(4x + 6) = 52\), so \(4x + 6 = 26\).
  3. 3 Solve \(4x = 20\), so \(x = 5\).
  4. 4 Answer the question Length \(= 3 \times 5 + 2 = 17\) cm. The width is \(5 + 4 = 9\) cm, and \(2(17 + 9) = 52\) as required.

Answer\(x = 5\), length = 17 cm

Solving inequalities

Inequalities are solved like equations, with one important difference.

  • The symbols

    \(<\) means less than, \(>\) means greater than, \(\leq\) means less than or equal to and \(\geq\) means greater than or equal to.

  • Solving

    \(3x + 2 < 14\). Subtract 2 to get \(3x < 12\), then divide by 3 to get \(x < 4\).

  • The negative rule

    If you multiply or divide both sides by a negative number, you must reverse the inequality sign. \(-2x > 6\) gives \(x < -3\).

  • Integer solutions

    List the whole numbers that fit. \(-2 \leq x < 4\) has the integer solutions \(-2, -1, 0, 1, 2, 3\), and 4 is not included.

A double inequality

Solve \(-3 < 2x + 1 \leq 9\) and list the integer values of \(x\) that satisfy it.

Show the solutionHide the solution
  1. 1 Subtract 1 from every part \(-4 < 2x \leq 8\).
  2. 2 Divide every part by 2 \(-2 < x \leq 4\).
  3. 3 List the integers \(x\) must be bigger than \(-2\) but can equal 4, so the integers are \(-1, 0, 1, 2, 3, 4\).

Answer\(-2 < x \leq 4\), so \(x = -1, 0, 1, 2, 3, 4\)

Equations and inequalities

Solving an equation

  • The solution is a particular value or values
  • The equals sign stays throughout
  • Check by substituting the answer back in

Solving an inequality

  • The solution is a range of values
  • The inequality sign stays, but reverses if you multiply or divide by a negative
  • Show the answer on a number line, using open and filled circles

A method for any linear equation

The same order of steps works for every linear equation.

  1. 1 Expand any brackets

    Multiply out so there are no brackets left.
  2. 2 Collect the unknowns

    Add or subtract so that every term in \(x\) is on one side and every number is on the other.
  3. 3 Simplify each side

    Collect like terms on each side.
  4. 4 Divide

    Divide both sides by the coefficient of \(x\).
  5. 5 Check

    Substitute your answer into the original equation.

Equations from consecutive numbers and ages

Word problems give you an equation when you choose a letter sensibly.

  • Consecutive integers

    Three consecutive integers can be written \(n\), \(n + 1\) and \(n + 2\), and their sum is \(3n + 3\).

  • Consecutive even numbers

    Write them as \(n\), \(n + 2\) and \(n + 4\), where \(n\) is even.

  • Ages

    Let the youngest person's age be \(x\), and write the others in terms of \(x\).

  • Check

    Put your answer back into the story to see whether it makes sense.

Consecutive numbers

Three consecutive whole numbers add up to 72. Find the three numbers.

Show the solutionHide the solution
  1. 1 Write the numbers Let the smallest be \(n\), so the numbers are \(n\), \(n + 1\) and \(n + 2\).
  2. 2 Make an equation \(n + (n + 1) + (n + 2) = 72\), so \(3n + 3 = 72\).
  3. 3 Solve \(3n = 69\), so \(n = 23\).
  4. 4 State the numbers 23, 24 and 25. Check: \(23 + 24 + 25 = 72\).

Answer23, 24 and 25

A fraction with an expression on top

Solve \(\dfrac{3x - 2}{4} = 4\).

Show the solutionHide the solution
  1. 1 Multiply both sides by 4 \(3x - 2 = 16\).
  2. 2 Add 2 to both sides \(3x = 18\).
  3. 3 Divide by 3 \(x = 6\).
  4. 4 Check \(\dfrac{3 \times 6 - 2}{4} = \dfrac{16}{4} = 4\).

Answer\(x = 6\)

Writing inequalities from words and number lines

Translating between words and symbols is often worth marks on its own.

  • Words to symbols

    "At least 5" means \(x \geq 5\). "More than 3" means \(x > 3\). "Fewer than 12" means \(x < 12\). "No more than 20" means \(x \leq 20\).

  • Number lines

    A filled circle at 2 with an arrow to the left means \(x \leq 2\).

  • Two limits

    "\(x\) is greater than \(-1\) and at most 3" is written \(-1 < x \leq 3\).

  • Integer lists

    Check each end value to see whether it is included.

An inequality with brackets and unknowns on both sides

Solve \(2(x + 3) \geq 3x - 1\).

Show the solutionHide the solution
  1. 1 Expand the bracket \(2x + 6 \geq 3x - 1\).
  2. 2 Collect the \(x\) terms on the side where they are bigger Subtract \(2x\) from both sides: \(6 \geq x - 1\).
  3. 3 Add 1 to both sides \(7 \geq x\).
  4. 4 Write with \(x\) first \(x \leq 7\). Collecting \(x\) on the right avoided dividing by a negative number.

Answer\(x \leq 7\)

Test yourself

  1. 1

    Solve \(x + 9 = 4\).

    Show answerHide answer

    \(x = -5\).

  2. 2

    Solve \(4x = 18\).

    Show answerHide answer

    \(x = 4.5\).

  3. 3

    Solve \(3x - 5 = 10\).

    Show answerHide answer

    \(x = 5\).

  4. 4

    Solve \(2(x + 1) = 10\).

    Show answerHide answer

    \(x = 4\).

  5. 5

    Solve \(\dfrac{x}{3} = 7\).

    Show answerHide answer

    \(x = 21\).

Exam technique: equations and inequalities

Layout is half the battle in equations, and it is easy to keep tidy.

  • One step per line

    Write each new line below the last, with the same operation applied to both sides.

  • Substitute to check

    Put your answer back in. It takes seconds and shows you whether you have made an error.

  • Define your letter

    In worded problems, say what \(x\) stands for before you write the equation.

  • Inequalities

    Remember to reverse the sign when you multiply or divide by a negative, and read the question to see whether integers are needed.

Summary and exam focus

  • Do the same to both sides, and undo the addition or subtraction before the multiplication or division.
  • With unknowns on both sides, collect the \(x\) terms on one side first, and expand brackets before you collect.
  • To form an equation, say what \(x\) stands for and use the fact in the question, such as a perimeter or an angle sum.
  • Solve an inequality like an equation, but reverse the sign when multiplying or dividing by a negative number.
  • A filled circle means included and an open circle means not included.

Exam focus

Solve \(4(x - 3) = 2x + 7\). (3 marks) (3 marks)

Expand the bracket first and write the new equation on its own line. Your answer may not be a whole number: \(x = 9.5\) is perfectly acceptable, so check by substituting rather than assuming something has gone wrong.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Boundary value
The value at which an inequality changes from true to false, such as 4 in \(x < 4\).
Equation
A statement that two expressions are equal, which is true only for certain values of the unknown.
Solve
Find the value or values of the unknown that make an equation or inequality true.
Inverse operation
The operation that undoes another, such as subtraction undoing addition.
Unknown
A letter that stands for a number you need to find.
Inequality
A statement that compares two expressions using \(<\), \(>\), \(\leq\) or \(\geq\).
Integer solutions
The whole-number values that satisfy an inequality.
Open circle
The symbol on a number line showing that an end value is not included.
Filled circle
The symbol on a number line showing that an end value is included.
Substitute
Replace a letter with a number.

Questions and answers

17 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Solve 2 marks Easier

Solve \(5x + 7 = 32\).

Mark scheme — 2 marks available

  • \(5x = 25\) or \(\dfrac{32 - 7}{5}\) — M1
  • \(x = 5\) — A1

Model answer

Subtract 7 from both sides to get \(5x = 25\), then divide by 5 to get \(x = 5\).

2. Exam question Solve 3 marks Core

Solve \(9 - 2x = 3x - 11\).

Mark scheme — 3 marks available

  • \(9 = 5x - 11\) or \(20 - 2x = 3x\) — M1
  • \(5x = 20\) — M1
  • \(x = 4\) — A1

Model answer

Add \(2x\) to both sides to get \(9 = 5x - 11\). Add 11 to get \(20 = 5x\). So \(x = 4\). Check: \(9 - 8 = 1\) and \(12 - 11 = 1\).

3. Exam question Solve 3 marks Core

Solve \(2(3x + 4) = 5(x + 2) + 3\).

Mark scheme — 3 marks available

  • Expands correctly: \(6x + 8\) and \(5x + 10\) — M1
  • \(6x + 8 = 5x + 13\) or \(x + 8 = 13\) — M1
  • \(x = 5\) — A1

Model answer

Expand both sides: \(6x + 8 = 5x + 10 + 3\), which is \(6x + 8 = 5x + 13\). Subtract \(5x\): \(x + 8 = 13\). So \(x = 5\).

4. Exam question Solve 3 marks Core

Solve \(\dfrac{3x + 1}{2} = x + 3\).

Mark scheme — 3 marks available

  • \(3x + 1 = 2(x + 3)\) or equivalent — M1
  • \(3x + 1 = 2x + 6\) — M1
  • \(x = 5\) — A1

Model answer

Multiply both sides by 2 to get \(3x + 1 = 2(x + 3)\), which is \(3x + 1 = 2x + 6\). Subtract \(2x\) and 1: \(x = 5\).

5. Exam question Work out 4 marks Core

Ben is \(x\) years old. His sister is 3 years older than Ben. Their mother is 4 times as old as Ben. The sum of their three ages is 69 years. Work out Ben's age and his mother's age.

Mark scheme — 4 marks available

  • \(x + 3\) and \(4x\) as expressions for the sister and mother — M1
  • \(x + (x + 3) + 4x = 69\) or equivalent — M1
  • Ben is 11 — A1
  • Mother is 44 — A1

Model answer

Ben is \(x\), his sister is \(x + 3\) and his mother is \(4x\). So \(x + (x + 3) + 4x = 69\), which gives \(6x + 3 = 69\), so \(6x = 66\) and \(x = 11\). Ben is 11 and his mother is \(4 \times 11 = 44\).

6. Exam question Solve 3 marks Stretch

Solve \(2 - 3x \leq 14\).

Mark scheme — 3 marks available

  • \(-3x \leq 12\) — M1
  • Divides by \(-3\) and reverses the sign — M1
  • \(x \geq -4\) — A1

Model answer

Subtract 2 from both sides to get \(-3x \leq 12\). Divide both sides by \(-3\) and reverse the inequality sign, giving \(x \geq -4\).

7. Exam question Solve 3 marks Core

(a) Solve \(4x + 3 > 15\). [2 marks] (b) Write down the smallest integer that satisfies \(4x + 3 > 15\). [1 mark]

Mark scheme — 3 marks available

  • (a) \(4x > 12\) — M1
  • (a) \(x > 3\) — A1
  • (b) 4 — B1

Model answer

(a) Subtract 3 to get \(4x > 12\), then divide by 4 to get \(x > 3\). (b) 3 is not included because the sign is \(>\), so the smallest integer is 4.

8. Multiple choice 1 mark Core

Three consecutive integers add up to 72. What is the smallest of them?

  1. A 24
  2. B 23 Correct
  3. C 25
  4. D 21

Why: Let the numbers be \(n\), \(n + 1\) and \(n + 2\). Then \(3n + 3 = 72\), so \(n = 23\).

9. Multiple choice 1 mark Core

Which inequality means "at most 20"?

  1. A \(x > 20\)
  2. B \(x \geq 20\)
  3. C \(x < 20\)
  4. D \(x \leq 20\) Correct

Why: "At most 20" means 20 or less, which is \(x \leq 20\).

10. Multiple choice 1 mark Easier

Solve \(2x + 7 = 19\).

  1. A \(x = 13\)
  2. B \(x = 12\)
  3. C \(x = 6\) Correct
  4. D \(x = 5\)

Why: Subtract 7 to get \(2x = 12\), then divide by 2.

11. Multiple choice 1 mark Core

Solve \(5x - 3 = 2x + 9\).

  1. A \(x = 6\)
  2. B \(x = 12\)
  3. C \(x = 2\)
  4. D \(x = 4\) Correct

Why: Subtract \(2x\) to get \(3x - 3 = 9\), add 3 to get \(3x = 12\), and divide by 3.

12. Multiple choice 1 mark Core

Solve \(3(x + 2) = 18\).

  1. A \(x = 16\)
  2. B \(x = 4\) Correct
  3. C \(x = 5\)
  4. D \(x = 8\)

Why: Expand to get \(3x + 6 = 18\), so \(3x = 12\) and \(x = 4\).

13. Multiple choice 1 mark Core

Solve \(\dfrac{x}{5} - 2 = 3\).

  1. A \(x = 5\)
  2. B \(x = 15\)
  3. C \(x = 1\)
  4. D \(x = 25\) Correct

Why: Add 2 to get \(\dfrac{x}{5} = 5\), then multiply both sides by 5.

14. Multiple choice 1 mark Stretch

Solve \(-2x > 6\).

  1. A \(x < 3\)
  2. B \(x > 3\)
  3. C \(x > -3\)
  4. D \(x < -3\) Correct

Why: Dividing by \(-2\) reverses the inequality sign, so \(x < -3\).

15. Multiple choice 1 mark Core

Which list shows all the integers that satisfy \(-2 \leq x < 2\)?

  1. A \(-2, -1, 0, 1\) Correct
  2. B \(-2, -1, 0, 1, 2\)
  3. C \(-1, 0, 1\)
  4. D \(-1, 0, 1, 2\)

Why: \(-2\) is included because of \(\leq\), but 2 is not included because of \(<\).

16. Multiple choice 1 mark Core

Which inequality is shown by an open circle at 5 with an arrow pointing to the right?

  1. A \(x > 5\) Correct
  2. B \(x < 5\)
  3. C \(x \geq 5\)
  4. D \(x \leq 5\)

Why: An open circle means 5 is not included, and the arrow to the right means larger numbers, so \(x > 5\).

17. Multiple choice 1 mark Core

I think of a number, double it and subtract 3. The answer is 11. Which equation represents this?

  1. A \(x - 6 = 11\)
  2. B \(2x - 3 = 11\) Correct
  3. C \(2(x - 3) = 11\)
  4. D \(2x + 3 = 11\)

Why: Doubling gives \(2x\) and subtracting 3 gives \(2x - 3\), which equals 11.