Pure Mathematics Practice: Equations and Inequalities

Original practice material

These four structured questions total 35 marks. They combine short technique parts with linked reasoning suitable for Paper 1 or Paper 2 Section A; they are not a complete paper.

Answers: Equations and Inequalities Answers.

Question 1: Quadratic and Rational Inequalities

(a) Solve

$$(x-2)(x+5)<0.$$

[3]

(b) Solve

$$\frac{x-4}{x+2}\ge 0.$$

State clearly whether each critical value is included. [4]

(c) Explain why multiplying a rational inequality by an expression containing $x$ without first determining its sign is unsafe. [2]

[Total: 9]

Question 2: Modulus Equations and Inequalities

(a) Solve $|2x-3|=5$. [3]

(b) Solve $|x+1|<4$. [3]

(c) Solve $|x-2|\ge |2x+1|$. [4]

[Total: 10]

Question 3: A System of Linear Equations

Solve the simultaneous equations

$$\{\begin{array}{l}2x-y+z=5,\\ x+2y-z=1,\\ 3x+y+2z=12.\end{array}$$

Check your solution in one of the original equations. [6]

Question 4: Parameter in an Inequality

Let

$$q(x)=x^2-2kx+9.$$

(a) Find the discriminant of $q(x)$. [2]

(b) Find the values of $k$ for which $q(x)>0$ for every real $x$. [4]

(c) For $k=4$, solve $q(x)\le 0$. [4]

[Total: 10]