Equations and Inequalities

In an interview you will be expected to know how to solve GCSE level equations and inequalities (though we don't mind if you make an occasional arithmetical or algebraical mistake!).  Some things to think about when dealing with equations and inequalities:

• Factorisation is your friend!  This includes both writing a polynomial as a product of brackets and expressing an integer as a product of prime factors.
• The factor theorem is not just for numbers.  For example if you substitute $a= - b$ into $a^3 + b^3$ you get $(-b)^3 +b^3=0$.  This means that $a+b$ is a factor of $a^3 + b^3$.  See this video for the sum of two cubes factorisation.
• Be careful about dividing by something that might be 0, as this can lead to "lost solutions".
• Be careful when squaring both sides as this can create "extra solutions".
• If trying to prove that $A>B$ it is often easier to show instead that $A-B>0$.  See this video for a classic example!
• If applying an "increasing function" to both sides of an inequality then the equality sign is preserved (for example, taking logs of both sides).  See this video for some more explanations of what does and does not preserve an inequality sign.
• Check that your answers make sense!

Interview question examples

Here are some questions for you to try.  Each one has a link to a video where we discuss the question and some possible methods which can be used to solve it.

Question 1: Solve the inequality $1+2x-x^2 > \dfrac 2 x$ where $x \neq 0$.

This question is the first part of 2001 STEP 1 Q1.  In the video below we discuss some of the possible methods.

Question 2: Find all the solutions to $\sqrt{x+3}=\sqrt{1+x} +\sqrt{1-x}$.  Now find all the solutions to  $\sqrt{x+3}=\sqrt{1+x} -\sqrt{1-x}$.  Can you explain the difference?

In the video below we discuss this problem, and in the discussion we use some graph sketching techniques, including finding the gradient by using the chain rule for differentiation.  Don't worry if you have not met that yet!

Question 3: I have a set of three numbers, $(a, b, c)$, which have the property that if we multiply any two together and add the third then we get the value 2.  What are the numbers?

This question is taken from the Nrich Problem "Always two", where there are also a couple of extension problems to try.  In the video below we talk about a possible solution method to the problem.  

Question 4a: Imagine you have three sticks, whose lengths are in a geometric progression.  For what values of the common ratio can the three sticks form a triangle?

Question 4b: A triangle has integer lengths, is right angled and has the property that the perimeter of the triangle is equal to the area of the triangle.  What are the possible values of the lengths of the triangle?

In the video below two of our undergraduate mentors tackle these questions as part of a mock interview.

More practice questions

Here are some more questions which involve solving equations or inequalities.   For STEP questions use the STEP database to find the question.

• "Symmetry, Substitution and Surprises" - a collection of problems from Nrich
• "Sums, Squares and Substantiation" - another collection of problems from Nrich
• 2013 STEP 1 Question 1.  This question uses substitutions to solve equations.  Part (i) is a hint for how to do part (ii).
• 2008 STEP 1 Question 3.  This question is about inequalities.  Remember it is easier to show that $A-B > 0$ than to show that $A>B$.
• 2000 STEP 1 Question 6. This question combines factorisation and sketching graphical inequalities.
• The STEP Support Programme foundation modules, especially assignments 1 to 6, assignment 10 and assignment 11.