About this chapter
Equations and inequalities is where Pure 1 starts to reward careful notation. The mathematics is often straightforward, but examiners regularly take marks for how an answer is written: a range stated the wrong way round, a missing boundary on a region, or roots copied from a calculator.
This chapter covers simultaneous equations with one linear and one quadratic equation, linear and quadratic inequalities, inequalities with x in the denominator, and shading and defining regions. The longer questions put these in context: a rectangle from its perimeter and diagonal, a region whose area fixes a constant, and the shortest distance from a point to a line.
The ten questions
- Q1 (6 marks): solve y − 3x + 2 = 0 and x² + y² = 20 simultaneously, pairing each x with its y.
- Q2 (7 marks): a linear inequality, a quadratic inequality, then the values satisfying both.
- Q3 (4 marks): solve 5 − 12/x > 2 by multiplying through by x², not by x.
- Q4 (6 marks): sketch y = 4 − x² and y = x + 2, find where they meet, and shade a region with solid and dashed boundaries.
- Q5 (5 marks): find where a parabola lies above a line, then write down where it does not.
- Q6 (7 marks): a line meets a curve at P and Q, found without a calculator, then the region they bound with the y-axis.
- Q7 (9 marks): a rectangle with perimeter 46 cm and diagonal 17 cm, then the lengths giving an area of at least 120 cm².
- Q8 (8 marks): the line y = kx + 3 and the curve y = 1/x: when they meet twice, then the exact surd meeting points for k = 2.
- Q9 (7 marks): a triangular region whose area of 24 fixes k, then inequalities defining it.
- Q10 (10 marks): distances from A(3, 0) to points on a line: an equation, an inequality, and the shortest distance by completing the square.
Key skills tested
Simultaneous equations. Rearrange the linear equation and substitute into the other, then pair each x with its y. For example, y = x + 1 and x² + y² = 13 give (2, 3) and (−3, −2).
Linear inequalities. Solve as an equation, but reverse the sign when multiplying or dividing by a negative. For example, 3 − 2x > 9 gives x < −3.
Quadratic inequalities. Find the critical values, sketch the curve and choose the region: inside gives a < x < b, outside gives x < a or x > b.
Combining inequalities. Use a number line to find values that satisfy both conditions, and never write a range such as −1 > x > 2.
x in the denominator. Multiply by x², which is always positive, rather than by x. For example, 4/x < 1 gives 4x < x², so x < 0 or x > 4.
Inequalities as graphs. f(x) > g(x) means the graph of f is above the graph of g.
Shading regions. A solid line means the boundary is included; a dashed line means it is not. For example, y > x² needs a dashed curve with the region above it.
Defining a region. Write one inequality for each boundary, including vertical ones such as x > 0.
Forming equations from a context. Turn each fact into an equation or inequality, and check that the answers make sense in the situation.
Worked example
Question 3 from this chapter. Find the set of values of x for which 5 − 12/x > 2, where x is not zero.
Multiply both sides by x², which is positive, so the inequality sign stays the same: 5x² − 12x > 2x². Collect the terms: 3x² − 12x > 0, which factorises as 3x(x−4) > 0.
The critical values are 0 and 4. The graph of 3x(x−4) is a U-shaped parabola, positive outside its roots, so the answer is x < 0 or x > 4.
Multiplying by x instead gives only x > 4, which misses every negative solution. In the mark scheme that approach scores nothing unless both signs of x are considered separately.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2023, Q1: a quadratic inequality under a calculator warning, like Q2(b) and Q5.
- June 2023, Q2: simultaneous equations from a rectangle, with the length longer than the width, as in Q7.
- June 2023, Q7: an area in terms of an unknown fixes a constant, then the region is defined, as in Q9.
- June 2024, Q6: a line meets a curve without a calculator, then a region needs an x boundary, as in Q6.
- June 2024, Q9: a point on a line at a given distance, found with Pythagoras, as in Q10.
- January 2023, Q10(a): combining two inequalities into one set of values, as in Q2(c).
Where marks are lost
- Notation for an outside region. In June 2023, writing the answer in the form −1/4 > x > 2 cost many students the final mark. Write x < −1/4 or x > 2.
- The forgotten x boundary. When defining a region in June 2024, the restriction on x was the inequality most often left out.
- Calculator roots dressed up as working. Writing a factorisation such as (x−8)(x+6) for 2x² − 4x − 96 = 0, which does not multiply back to the equation, lost two marks in June 2024.
- Ignoring a stated condition. In June 2023 the final mark went when two pairs of answers were left, although the question said the length was greater than the width.
Common questions
Why multiply by x² and not by x?
x could be negative, and multiplying by a negative number reverses the inequality. x² is always positive, so the sign stays the same whichever x you have.
Should I use "and" or "or"?
Use "or" for two separate ranges, such as x < 0 or x > 4. "And" describes values that satisfy both at once, and x < 0 and x > 4 describes no values at all.
Solid or dashed boundary?
Solid when the boundary is included, for ≤ or ≥; dashed when it is not, for < or >.
Where this chapter leads
- Pure 1 Chapter 5: Straight-line Graphs: where lines meet, distances and areas.
- Pure 1 Chapter 9: Integration: two conditions give simultaneous equations for two unknown constants.
- Pure 2 Chapter 2: Coordinate Geometry: a line meeting a circle, solved with simultaneous equations.