Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Algebra
Practise Inequalities. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Solving inequalities such as \(3x - 4 \lt 11\), showing solutions on a number line and listing integer values (both tiers). Higher also solves quadratic inequalities, uses set notation and shows regions on a graph.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
3
Write an inequality from a number lineAn open circle means the value is not included (\(\lt\) or \(\gt\)); a filled circle means it is included (\(\le\) or \(\ge\)).
4
List integers that satisfy an inequalitye.g. \(-2 \lt n \le 3\) gives \(n = -1, 0, 1, 2, 3\).
4
Solve a linear inequalitySolve it like an equation, e.g. \(4x + 1 \gt 13\) gives \(x \gt 3\).
5
Solve a double inequalityDo the same to all three parts, e.g. \(-3 \le 2x + 1 \lt 9\) gives \(-2 \le x \lt 4\).
5
Reverse the sign when dividing by a negativee.g. \(-2x \gt -4\) gives \(x \lt 2\).
Notes
Symbols and number lines
\(\lt\) less than, \(\gt\) greater than, \(\le\) less than or equal to, \(\ge\) greater than or equal to.
On a number line, an open (empty) circle means the value is not included; a filled circle means it is included.
\(-1 \lt x \le 4\): open circle at −1, filled circle at 4, and a line joining them.
Integers are whole numbers, including 0 and negative whole numbers. For \(-1 \lt n \le 4\), \(n = 0, 1, 2, 3, 4\).
Solving linear inequalities
Solve like an equation: add, subtract, multiply or divide both sides by the same number.
If you multiply or divide by a negative number, reverse the inequality sign: \(-3x \lt 12\) gives \(x \gt -4\).
You can avoid dividing by a negative by moving the x term: \(7 - 2x \ge 1\) gives \(6 \ge 2x\), so \(x \le 3\).
Double inequalities: do the same to all three parts. \(5 \lt 3x - 1 \le 14\) gives \(6 \lt 3x \le 15\), so \(2 \lt x \le 5\).
Cheatsheet
\(\lt\) and \(\gt\): open circle. \(\le\) and \(\ge\): filled circle
Multiply or divide by a negative: reverse the sign
Double inequality: do the same to all three parts
Integers include 0 and negative whole numbers
How to answer each type of question
Solve and show on a number line
3 marks4
Solve as you would an equation, keeping the inequality sign.
Draw the correct circle (open or filled) at the boundary value.
Draw the line or arrow in the right direction.
Example. (a) Solve \(4x + 3 \gt 19\). (b) Show your answer on a number line.
Show the model answer
(a) \(4x \gt 16\) M1, \(x \gt 4\) A1 (b) Open circle at 4 with an arrow pointing to the right B1
List the integers
2 marks5
Solve the double inequality for n.
List every integer in the range, checking whether each end value is included.
Example. n is an integer and \(-3 \lt 2n \le 8\). Write down all the possible values of n.
Show the model answer
\(-1.5 \lt n \le 4\) M1 −1, 0, 1, 2, 3, 4 A1
Shortcuts and memory tricks
Check with a number: for \(x \gt 4\), try \(x = 5\) in the original inequality (it should work) and \(x = 3\) (it should not).
The wide, open side of \(\lt\) or \(\gt\) always faces the bigger value.
Higher: for a quadratic inequality, always sketch the parabola; it shows at once which part you need.
Where marks are lost
Writing an equals sign in the final answer: the answer to an inequality is an inequality, e.g. \(x \gt 4\).
Forgetting to reverse the sign when multiplying or dividing by a negative number.
Missing 0 or the negative values when listing integers, or including an end value that is not allowed.
Higher: writing 'outside' solutions as one double inequality such as \(-4 \ge x \ge 3\); write \(x \le -4\) or \(x \ge 3\).
Exam technique
Read whether the question wants integers or all possible values of x.
Keep the inequality sign on every line of working rather than switching to = and back.
On number lines, make open and filled circles clearly different.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
Solve \(3(x + 6) \ge 24\)[2]
Show the answer and mark scheme
Answer: \(x \ge 2\)
M1 for \(3x + 18 \ge 24\) or \(x + 6 \ge 8\)
A1 for \(x \ge 2\) oe
Worked solution: \(x + 6 \ge 8\), so \(x \ge 2\)
Question 2Medium4 marks
Two taxi firms charge for a journey of \(d\) miles. Firm A: a fixed charge of £3 plus £1.40 per mile. Firm B: £1.80 per mile, with no fixed charge.
(a) Write down an inequality, in terms of \(d\), for the journeys where Firm A is cheaper than Firm B. Solve your inequality.[3]
(b) Kim says, 'Firm B has no fixed charge, so it is always cheaper. I will use Firm B for my 9-mile journey.' Is Kim correct? Give a reason for your answer.[1]