Common Mistakes with Inequalities in GCSE Maths

The algebra can be right and the answer still wrong

You finish an inequalities question, recognise every algebraic step, and still lose marks. The common mistakes with inequalities in GCSE Maths usually involve reversing a sign, including the wrong endpoint, or giving a boundary instead of the full solution. The fix is to check three things separately: your algebra, the direction of the inequality, and the form of your answer.

Imagine a student reviewing a marked practice sheet. Rather than writing “careless” beside every correction, they identify exactly what went wrong. That makes the next revision session useful: each error becomes a rule they can practise.

The worked examples below follow that approach. They cover linear inequalities, number lines and integer solutions, with practice questions to check whether the corrections have stuck.

Your inequalities checklist

Before submitting an answer, ask:

• Meaning: does the symbol match “greater than” or “less than”?
• Operations: did I reverse the inequality when multiplying or dividing by a negative number?
• Endpoints: is the boundary included or excluded?
• Instructions: does the question want an inequality, a number line or integer values?
• Check: does a value from my answer satisfy the original inequality?

The MathsGenie inequalities topic page brings together lessons and practice for these skills. First, though, it helps to understand why each mistake happens.

Common mistakes with inequalities in GCSE Maths

Treating an inequality like an equation

The mistake: solving an inequality but replacing the final sign with an equals sign.

This happens because the algebra feels familiar. However, an equation asks for values that make two expressions equal; an inequality describes values that satisfy a comparison.

Solve:

$$2x+3<11$$

Subtract the same amount from both sides:

$$2x<8$$

Divide both sides by a positive number:

$$x<4$$

The answer is not $x=4$. In fact, substituting $x=4$ gives $11<11$, which is false. Every value below $4$ works, including non-integer values.

Repair habit: carry the inequality symbol through every line. Do not replace it with an equals sign just because you are doing algebra.

Forgetting to reverse the sign

The mistake: dividing by a negative number without reversing the inequality.

Solve:

$$-3x\le 12$$

Dividing both sides by $-3$ reverses the comparison:

$$x\ge -4$$

Why? Multiplying or dividing by a negative number reverses numerical order. Although $2<5$, multiplying both values by $-1$ gives $-2>-5$.

Check the solution using $x=0$:

$$-3\times 0\le 12$$

This is true, and $0$ belongs to the solution $x\ge -4$.

Repair habit: whenever you multiply or divide both sides by a negative number, write “reverse sign” beside that step until the habit becomes automatic.

Reversing the sign whenever a negative appears

The mistake: applying the previous rule too often.

A negative number appearing in a question does not itself reverse the sign. Neither does adding or subtracting a negative number.

For example:

$$x-5>-8$$

Add $5$ to both sides:

$$x>-3$$

The inequality direction stays unchanged. The rule concerns the operation you perform, not whether the question contains negative numbers.

If choosing inverse operations is the underlying difficulty, the solving equations revision page covers the algebra shared by equations and inequalities.

Changing only part of a double inequality

The mistake: performing an operation on two parts of a three-part inequality.

Solve:

$$-2<3x+1\le 10$$

Subtract $1$ from all three parts:

$$-3<3x\le 9$$

Divide all three parts by $3$:

$$-1<x\le 3$$

The expression in the middle must satisfy both restrictions. Changing only one boundary breaks that relationship.

Repair habit: write three aligned parts on each line, then check that every part received the same operation.

Use the inequalities lessons alongside the inequalities Question Bank to practise the particular step that keeps catching you out.

A stick-figure student brings an enormous eraser to revision as a humorous form of error control.

Number lines: two decisions, not one

Our hypothetical student might now have correct algebra but still draw the wrong answer. A number line introduces two separate decisions: whether the boundary belongs to the solution, and which direction contains the solutions.

Using the wrong endpoint

For $x<3$, draw an open circle at $3$. The boundary is excluded.

For $x\le 3$, draw a filled circle at $3$. The boundary is included.

The same distinction applies to greater-than inequalities. Strict inequalities, $<$ and $>$, exclude the boundary. The symbols $\le$ and $\ge$ include it.

Repair habit: decide “included or excluded?” before drawing the circle. The words give the symbol a meaning rather than leaving it as something to memorise.

Drawing in the wrong direction

The statement $4>x$ means exactly the same as $x<4$. Both describe values below $4$, so the number line extends left.

A common error is to see the greater-than symbol in $4>x$ and automatically draw right. That ignores which side contains the variable.

Repair habit: rewrite the statement with the variable on the left first. Then test a number from the direction you intend to draw. For $x<4$, choosing $x=3$ confirms that the left-hand side of the number line is correct.

Integer solutions: read the final instruction

The mistake: giving a continuous range when the question asks for integers, or including an excluded endpoint.

Suppose you must list the integers satisfying:

$$-2<n\le 3$$

The complete list is:

$$-1,0,1,2,3$$

The lower boundary, $-2$, is excluded. The upper boundary, $3$, is included. Zero and negative whole numbers are integers too.

There is another trap when a boundary is fractional. If $n<\frac{7}{2}$ and $n$ is an integer, the largest possible value is $3$. Do not round the boundary to the nearest integer: decide which integers genuinely satisfy the inequality.

Repair habit: underline “integer”, “smallest” or “largest” before answering. Those words determine the required form of your response.

Exam-style practice

Try these without looking back at the examples. Show your algebra, even when you can see the answer mentally.

• Sign reversal: solve $7-2x>15$.
• Double inequality: solve $-5\le 2x+1<7$.
• Integer values: list the integers satisfying $-\frac{3}{2}<n\le 2$.
• Number line: describe how to represent $2\ge x$.

Answers and checks

For the first question:

$$-2x>8$$

Dividing by $-2$ reverses the sign:

$$x<-4$$

For the second question, subtract $1$ from all three parts, then divide by $2$:

$$-6\le 2x<6$$ $$-3\le x<3$$

The integer values in the third question are:

$$-1,0,1,2$$

For the number line, rewrite the statement as $x\le 2$. Draw a filled circle at $2$ and extend the line left.

If your answer differs, identify the first incorrect step rather than simply copying the solution. MathsGenie's inequalities video lessons provide another explanation when a written correction is not enough.

Turn corrections into reliable marks

Your goal is not merely to recognise an inequalities mistake after marking; it is to catch it while answering. The gap closes when you revisit the exact skill, attempt fresh questions and check whether the same error returns. Start with the inequalities Lessons, then use the Question Bank to practise the correction you need.