TMUA Inequalities

Manipulating inequalities is an essential skill to master in order to succeed in TMUA. Even though they are rarely tested directly, they appear in all sorts of guises, from inequality equations to graphical inequalities and areas they enclose, to logic and reasoning questions involving inequalities (especially with a few negative numbers thrown in) - particularly, true/false, necessary and sufficient and 'if then' type logic. 

For those so interested, we offer both one to one TMUA coaching on inequalities, as well as online group coaching via Zoom - 'TMUA Prerequisite Maths One' is Session #1 of 20 and covers the basics of inequalities, and other sessions build on this base when dealing with such topics as graphs, polynomials and logic and reasoning. Check our TMUA Group Coaching Schedule page for dates when these sessions are available - details of what we cover in each group session can be found on our TMUA Coaching Sessions Overview page. For more general information on our online sessions check out our  'Online Coaching FAQ'.

Feel free to contact us if you require more information regarding any of the above.

Free Revision Resources

The below list of questions links to short videos (teasers) on our YouTube channel which directly or indirectly test elements of what is required to succeed in answering TMUA questions on inequalities - they also provide a detailed solution to the question posed, together with tips and tricks and are arranged roughly in increasing order of difficulty. All the videos are taken from the Gresty Academy YouTube podcast 'A Crash Course in TMUA Must Know Maths' which contains hundreds of TMUA style questions and is great for revision.

• Teaser #431 - tests basic understanding of what constitutes a 'counterexample to 'If X then Y' using inequalities - Fairly Mild
• Teaser #429 - tests inequality knowledge in relation to necessary and sufficient logic - Standard
• Teaser #474 - tests ability to solve two modulus inequalities and deduce a maximum of mod(y-x) - Standard
• Teaser #493 - tests understanding of the various components of logarithms and exponents and their allowed values - Standard
• Teaser #437 - tests which statements are necessary for x>2 to be true - Standard
• Teaser #283 - tests ability to manipulate correctly an algebraic inequality - Standard
• Teaser #453 - tests how to solve inequalities to find maximum of logs with bases more and less than 1 - Medium
• Teaser #476 - tests how to solve an inequality involving a polynomial over a polynomial - Medium
• Teaser #501 - tests understanding of a well known functional inequality - Medium
• Teaser #424 - tests ability to find fraction of an interval where sin(3x)>=1/2 - Medium
• Teaser #422 - tests ability to solve modulus equation by splitting into regions - Medium
• Teaser #363 - tests understanding of how inequalities are represented on a graph - Medium
• Teaser #307 - tests manipulation of an algebraic inequality which is often calculated incorrectly - Medium
• Teaser #496 - tests criteria for which a triangle is described uniquely - Medium
• Teaser #12 - tests ability to find domain of a log of a factorised polynomial - Medium
• Teaser #393 - tests understanding of how inequalities are represented on a graph - Medium
• Teaser #494 - tests understanding of how to find positive solutions to a modulus equation - Medium
• Teaser #470 - tests sufficiency of various inequality and modulus statements, often using counterexamples to disprove - Medium
• Teaser #486 - tests recognition of a well known functional inequality - Medium
• Teaser #90 - tests understanding of compound interest calculations - (the calculation itself is beyond TMUA) - Medium
• Teaser #488 - tests understanding of log bases and their respective graphs and putting them in increasing order - Medium
• Teaser #106 - tests understanding of how to calculate compound interest (the final calculation is beyond TMUA) - Medium
• Teaser #190 - tests ability to arrange surds in increasing order using a little known inequality - Medium
• Teaser #450 - tests which logic inequalities are necessary for y>3 to be true - Medium
• Teaser #333 - tests understanding of how inequalities are represented on a graph - Medium
• Teaser #471 - tests which function provides a counterexample to an integral statement - Medium
• Teaser #498 - tests how to establish when both axes are tangent to a circle - Medium
• Teaser #427 - tests ability to arrange various expressions in increasing order by clever approximations - Medium
• Teaser #419 - tests understanding of an inequality and how it affects possible values of cos theta - Medium
• Teaser #464 - tests how to solve a horrific looking trig inequality by reducing it to a much simpler one - Quite Tough
• Teaser #504 - tests advanced manipulation of two log(log) functions to solve an inequality - Quite Tough
• Teaser #499 - tests how to establish which of six statements is true, given that exactly one is - Quite Tough
• Teaser #503 - tests understanding of how to find when an integral approximation of y=sin^2(x) underestimates the area - Quite Tough
• Teaser #444 - tests ability to solve two trig inequalities and find the fraction of a range satisfying both - Quite Tough
• Teaser #298 - tests ability to find a range of values where there is no common tangent between ln(x) and a cubic- Quite Tough
• Teaser #491 - tests understanding of all steps required  to correctly solve a log = log equation - Tough

TMUA Past Questions involving Inequalities: SP p1 q4; SP p1 q16; SP p2 q15; 2016 p1 q17; 2016 p1 q19; 2016 p2 q8; 2016 p2 q10; 2016 p2 q12; 2016 p2 q15; 2016 p2 q18; 2017 p1 q5; 2017 p1 q8; 2017 p1 q16; 2017 p1 q17; 2017 p1 q19; 2017 p1 q20; 2017 p2 q11; 2017 p2 q12; 2017 p2 q13; 2017 p2 q19; 2018 p1 q9; 2018 p1 q10; 2018 p1 q12; 2018 p2 q14; 2018 p2 q18; 2018 p2 q19; 2018 p2 q20; 2019 p1 q2; 2019 p1 q8; 2019 p1 q13; 2019 p1 q17; 2019 p1 q20; 2019 p2 q8; 2019 p2 q12; 2019 p2 q16; 2019 p2 q17; 2019 p2 q18; 2020 p1 q3; 2020 p1 q8; 2020 p1 q17; 2020 p2 q1; 2020 p2 q10; 2020 p2 q12; 2020 p2 q13; 2020 p2 q18; 2020 p2 q20; 2021 p1 q7; 2021 p1 q19; 2021 p2 q11; 2021 p2 q12; 2021 p2 q13; 2021 p2 q14; 2021 p2 q17; 2021 p2 q18; 2022 p1 q2; 2022 p1 q17; 2022 p1 q19; 2022 p2 q9; 2022 p2 q12; 2022 p2 q14; 2022 p2 q16; 2023 p1 q2; 2023 p1 q14; 2023 p1 q18; 2023 p2 q7; 2023 p2 q10; 2023 p2 q13; 2023 p2 q18; 2023 p2 q20

Practice makes perfect - good luck in your studies!