Linear Equations and Inequalities Worked ExamplesIGCSE Maths 0580

Moving the $x$ terms to the side with the bigger coefficient keeps them positive and avoids a sign error. Even on the simplest equations, the June 2025 examiner report recorded "a method error of subtracting rather than adding" the constant — here $2x - 7 = 9$ needs $+7$ on both sides. Substituting back takes one line and catches it.

Multiply the right-hand side by $12$ as well — leaving it as $1$ is a common slip. And $-3 \times -2 = +6$, not $-6$. The March 2026 examiner report noted sign errors when candidates collected the $x$ terms on one side after clearing the fractions, so take each step on its own line.

The reversal applies to multiplying or dividing by a negative, not to adding or subtracting one. Negative $x$ terms cause slips in equations too: the March 2026 examiner report found that many candidates made slips dealing with the negatives when solving an equation with a $-6x$ term. If you prefer to avoid it, add $2x$ to both sides instead: $5 < 11 + 2x$, then $-6 < 2x$, giving $x > -3$ with no reversal at all.

Part (b) turns on the inequality signs: $-2$ is included and $3$ is not. The March 2026 examiner report found that some candidates gave only three of four correct integers, with the included negative end value a common omission, and that "some candidates did not realise that 0 was an integer."

The June 2025 examiner report found that candidates "had more problems accessing questions where they were required to interpret a context or form an equation to solve a problem". Define your variable in writing first ('let $w$ be the width'), then write every length in terms of it. Answer both parts of the question; finding $w$ and stopping is a common way to drop the final mark.

Context decides the rounding, not the digit. Here $17.5$ rounds down to $17$ because exceeding the budget is not allowed — rounding to $18$ by habit loses the final mark. The March 2025 examiner report advised candidates "to check their answer makes sense for the context of the problem"; the check in step 5 does exactly that.

A common error is using $180^{\circ}$, the angle sum of a triangle. Working out all four angles and checking they add to $360^{\circ}$ catches it. In (b), $x = 33$ is not an angle, so giving it as the answer earns nothing.

The March 2025 examiner report noted that "some mixed up the meaning of the unshaded circle and the shaded circle". Here that gives $-2 \leqslant x < 4$, which loses a mark in (a) and changes (c) to $3$ only. In (b), candidates who subtract $2x$ and then divide by $-5$ often forget to reverse the sign. Keeping the $x$ term positive avoids the problem.

Treat each pair of sides as its own linear inequality and check all three. One pair gives nothing new, and saying so is part of a complete answer. In (c), the lower limit still comes from (a): listing $1, 2, \ldots, 6$ ignores the fact that the triangle cannot be drawn for $x \leqslant 2.75$.

The June 2025 examiner report found that most candidates chose to expand the bracket first, occasionally forgetting to multiply the second term by the number outside, and that some subtracted the constant from both sides instead of adding it. A fractional answer is fine — do not round it to $1.83$, which is not exact. Dividing by $3$ first ($4y - 5 = \frac{7}{3}$) also works but brings fractions in earlier.

The June 2025 examiner report found that "a significant number of candidates struggled to set up a correct equation" in a ticket context like this, and that a common error in solving was subtracting instead of adding. Write the child price as $a - 2.75$ in terms of the same letter, so there is only one unknown. Give money to the cent and answer both parts.

The June 2025 examiner report noted that "the most common errors were to calculate a mean of the test marks in the table" — the values in the top row — instead of using the frequencies. The mean of $0, 1, 2, 3$ is $1.5$ and has nothing to do with $x$. Remember the $1.6$ multiplies the whole of $18 + x$.

The June 2025 examiner report noted errors from forgetting to halve the total area when working with one trapezium, and from omitting brackets. Without the bracket, $\tfrac{1}{2}x + 12 \times 4.5$ is a different expression. Check: $2 \times \tfrac{1}{2}(8.4 + 12) \times 4.5 = 91.8$. ✓ The answer is exact, so give $8.4$, not a rounded value.