Linear equations and inequalities – Year 10 GCSE maths test answers to download as PDF

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1. Answer to Question 1 (4 marks)

   1. \(2x=12\) (1), so \(x=\) 6 (1) Subtract \(3x\) and 8 from both sides, then divide by 2. Check: \(30+8=38\) and \(18+20=38\). The solution is \(x=6\).
   2. \(3x-2=20\) (1), \(x=\) \(\dfrac{22}{3}\) (7.33…) (1) Multiply both sides by 4, add 2, then divide by 3. The solution is \(x=\tfrac{22}{3}\).

   Check (b) by substitution: \(3\times\tfrac{22}{3}-2=20\) and \(20\div4=5\). The exact fraction \(\tfrac{22}{3}\) or \(7\tfrac13\) is preferred; a rounded decimal such as 7.3 loses the accuracy mark. In (a), collecting the \(x\) terms on the side with the larger coefficient avoids negatives.

2. Answer to Question 2 (4 marks)

   1. First: \(45+28\times5=185\). Second: \(20+33\times5=185\) (1, 1). Both plumbers charge £185 for 5 hours, so they charge the same.
   2. \(45+28h\lt 20+33h\) (1), so \(25\lt 5h\) and \(h>5\) (1). Subtract \(28h\) and 20 from both sides, then divide by 5. The first plumber is cheaper for jobs longer than 5 hours.

   Check (b) with a value: for 6 hours the first plumber charges £213 and the second £218, so the first is indeed cheaper beyond 5 hours.

3. Answer to Question 3 (4 marks)

   1. \(4x\le20\) (1), so \(x\le5\) (1) Add 7 to both sides, then divide by 4. The solution is \(x\le5\).
   2. \(-6\lt 3x\lt 9\) so \(-2\lt x\lt 3\) (1). Integers: −1, 0, 1, 2 (1) Subtract 1 from all three parts, then divide by 3: \(-2\lt x\lt 3\). The integers are \(-1\), 0, 1 and 2.

   In (b), dividing by a positive number keeps the inequality signs the same way round, and \(-2\) and 3 are not solutions.

4. Answer to Question 4 (4 marks)

   1. \(80+6g\le350\) (1) The total cost is the fixed £80 plus £6 for each guest, and it must not be more than £350.
   2. \(6g\le270\) (1), \(g\le45\) (1) Subtract 80 from both sides, then divide by 6.
   3. 45 guests (1) The greatest whole number that satisfies \(g\le45\) is 45. Check: \(80+6\times45=350\).

   In (c), a guest cannot be a fraction, and here 45 guests use exactly the whole budget of £350, so 46 guests would cost £356 and break the budget.

5. Answer to Question 5 (4 marks)

   1. \((x-5)(x+2)\lt 0\) (1), roots \(-2\) and \(5\) (1), \(-2\lt x\lt 5\) (1) The graph of \(y=(x-5)(x+2)\) is a U-shape crossing the axis at \(-2\) and 5. It is below the axis between the roots, so \(-2\lt x\lt 5\).
   2. −1, 0, 1, 2, 3, 4 (1) The end values \(-2\) and 5 are not included because the inequality is strict.

   Check with \(x=0\): \(-10\lt 0\), true.

Total : 20 marks