2.2 Equations and inequalities
Equations and inequalities
Everything on equations and inequalities that can come up in the WJEC GCSE Mathematics and Numeracy, in Unit 2 and Unit 3. Each skill below has the method, the mistake that costs the most marks, a worked example, and free practice marked the way WJEC marks it.
Practise equations and inequalities free
A worked example
Two-step equation · [2 marks]
Solve the equation 6x + 12 = 18.
- M1Subtracts 12 on both sides: 6x = 6
- A1x = 1
The exam skills (14)
Two-step equation
- Undo the + or − first by doing the opposite to both sides.
- Then undo the × by dividing both sides.
- Check by substituting.
Watch out: Dividing first. Undo in reverse order.
Inequality
- Solve it like an equation.
- Keep the inequality sign in every line.
- If you multiply or divide by a negative, flip the sign.
Watch out: Writing = in the middle of the working.
Unknowns on both sides
- Get the x terms on the side with the bigger number of x.
- Get the numbers on the other side.
- Divide.
Watch out: Moving a term without changing its sign.
Equation with brackets
- Expand the bracket.
- Collect terms.
- Divide to get x.
Watch out: Only multiplying the first term in the bracket.
Integers in an inequality
- Solve each side of the double inequality (do the same to all three parts).
- Look carefully at < or ≤ at each end.
- List the whole numbers that fit.
Watch out: Including an end value when the sign is <.
Quadratic equation
- Make sure it equals 0.
- Factorise.
- Each bracket = 0 gives a solution. Flip the sign.
Watch out: Writing x = 9 from (x + 9) = 0. It is x = −9.
Simultaneous equations
- Make the x (or y) numbers the same in both equations.
- Same signs subtract, different signs add.
- Solve, then substitute back to find the other letter.
Watch out: Adding when the signs are the same.
Forming an equation
- Write the equation from the fact: angles in a triangle = 180, perimeter = …
- Solve for x.
- Substitute to answer what was actually asked.
Watch out: Stopping at x.
Forming an inequality
- Write each quantity in terms of x.
- Add them and use the inequality sign from the question.
- Solve; then choose the whole number that fits (the next one up for >).
Watch out: Rounding down. If x > 50.3, the smallest whole number is 51.
Brackets on both sides
- Expand both brackets.
- Collect x terms on one side, numbers on the other.
- Divide.
Watch out: Sign errors when expanding a negative.
Fractions on both sides
- Multiply both sides by both denominators (cross-multiply).
- Put brackets round each top.
- Expand, collect and solve.
Watch out: Not putting brackets round the tops.
Quadratic formula
- Write down a, b and c (including signs).
- x = (−b ± √(b² − 4ac)) ÷ 2a. Work out b² − 4ac first.
- Do + and − to get two answers; round at the end.
Watch out: Squaring a negative b wrongly. (−9)² = 81.
Quadratic inequality
- Solve the quadratic = 0 to find the critical values.
- Sketch the U-shaped curve through them.
- < 0 is between the values; > 0 is outside them.
Watch out: Giving two separate inequalities when it should be one in between.
Trial and improvement
- Try a value, work out the left side and compare with the target.
- Narrow it down to two 1 d.p. values either side of the solution.
- Try the halfway value (e.g. 2.35) to decide which way to round.
Watch out: Stopping before testing the halfway value.
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