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Test of Mathematics for University Admission (TMUA) Equations and Inequalities Syllabus
The Equations and Inequalities syllabus for Test of Mathematics for University Admission (TMUA) in full: 3 chapters, 10 topics and 20 sub-topics, every one listed below, plus 50 Equations and Inequalities flashcards with answers.
Equations and Inequalities syllabus: all 3 chapters
Expand any chapter to see its topics and sub-topics.
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Linear and Simultaneous Equations
3 topics- Solving Linear Equations
- Rearranging and changing the subject
- Equations with fractions
- Simultaneous Equations
- Linear-linear systems
- Linear-quadratic systems
- Geometric Interpretation
- Intersection of lines and curves
- Number of solutions
- Solving Linear Equations
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Quadratic Equations and the Discriminant
3 topics- Solving Quadratics
- Factorisation method
- Quadratic formula
- Completing the square
- The Discriminant
- Nature of roots
- Conditions for real or repeated roots
- Equations Reducible to Quadratics
- Disguised quadratics in u-substitution
- Quadratics in surds and exponentials
- Solving Quadratics
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Inequalities
4 topics- Linear Inequalities
- Single and combined inequalities
- Inequalities with negative multipliers
- Quadratic Inequalities
- Sign diagrams and sketch method
- Critical values and intervals
- Inequalities with Algebraic Fractions
- Sign analysis of rational expressions
- The Modulus Function
- Solving modulus equations
- Solving modulus inequalities
- Linear Inequalities
Test of Mathematics for University Admission (TMUA) Equations and Inequalities flashcards — 50 questions with answers
What is the general process for solving a linear equation in one variable?
Collect all variable terms on one side and constants on the other (using inverse operations applied equally to both sides), simplify to the form $ax = b$, then divide by the coefficient to get $x = \frac{b}{a}$ (provided $a \neq 0$).
When does the linear equation $ax = b$ have (i) a unique solution, (ii) no solution, (iii) infinitely many solutions?
(i) Unique solution $x=\frac{b}{a}$ when $a \neq 0$; (ii) no solution when $a = 0$ and $b \neq 0$; (iii) infinitely many solutions when $a = 0$ and $b = 0$.
How do you solve a linear equation containing fractions, e.g. $\frac{x}{3} + \frac{x}{4} = 7$?
Multiply every term by the lowest common multiple of the denominators (here $12$) to clear fractions: $4x + 3x = 84$, giving $7x = 84$, so $x = 12$.
What does it mean to solve a pair of simultaneous equations?
To find the value(s) of the unknowns that satisfy all the equations at the same time; geometrically, the coordinates of the point(s) where the graphs intersect.
Describe the elimination method for solving two linear simultaneous equations.
Scale one or both equations so that the coefficients of one variable are equal in magnitude, then add or subtract the equations to eliminate that variable, solve for the remaining variable, and back-substitute to find the other.
Describe the substitution method for solving simultaneous equations.
Rearrange one equation to make one variable the subject, substitute that expression into the other equation to get a single-variable equation, solve it, then back-substitute to find the remaining variable.
Which method is generally preferred for a linear–quadratic pair of simultaneous equations, and why?
Substitution: rearrange the linear equation for one variable and substitute into the quadratic, producing a single quadratic equation in one variable to solve.
For two linear equations, what determines whether they have one solution, no solution, or infinitely many solutions?
One solution: the lines have different gradients (intersect once). No solution: same gradient, different intercepts (parallel, distinct). Infinitely many: same gradient and same intercept (identical lines).
Geometrically, what does the solution of a linear–quadratic system represent, and how many solutions are possible?
The intersection points of a line and a parabola (or other conic). There can be $2$ solutions (line cuts the curve), $1$ solution (line is tangent), or $0$ solutions (line misses the curve).
How does the discriminant of the resulting quadratic tell you the geometric relationship between a line and a curve?
After substituting, the quadratic's discriminant $b^{2}-4ac$ gives: $>0$ two intersection points (secant), $=0$ one point (tangent), $<0$ no intersection.
State the quadratic formula for solving $ax^{2}+bx+c=0$.
$$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}, \quad a \neq 0.$$
What is the method of completing the square for $x^{2}+bx+c$?
Write $x^{2}+bx+c = \left(x+\frac{b}{2}\right)^{2} - \left(\frac{b}{2}\right)^{2} + c$, halving the coefficient of $x$ and subtracting the square of that half.
How do you solve a quadratic equation by factorising?
Write it as $ax^{2}+bx+c=0$, factorise into $(px+q)(rx+s)=0$, then set each factor to zero (using the null factor law) and solve, giving $x=-\frac{q}{p}$ or $x=-\frac{s}{r}$.
For $ax^{2}+bx+c=0$ with roots $\alpha$ and $\beta$, what are the sum and product of the roots?
$$\alpha+\beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}.$$
What is the formula for the discriminant of $ax^{2}+bx+c=0$, and what does it determine?
The discriminant is $\Delta = b^{2}-4ac$. It determines the number and nature of the real roots of the quadratic.
Classify the roots of a quadratic according to the sign of the discriminant $b^{2}-4ac$.
$b^{2}-4ac>0$: two distinct real roots. $b^{2}-4ac=0$: one repeated (equal) real root. $b^{2}-4ac<0$: no real roots (two complex conjugate roots).
What condition on the discriminant guarantees a quadratic has two distinct real roots?
$b^{2}-4ac > 0$.
What does $b^{2}-4ac=0$ tell you about a parabola $y=ax^{2}+bx+c$ and the $x$-axis?
The parabola touches the $x$-axis at exactly one point; the $x$-axis is tangent to it (a repeated root).
If a quadratic $ax^{2}+bx+c$ is a perfect square, what is the value of its discriminant?
$b^{2}-4ac = 0$, since a perfect square has a repeated root.
For what values of $k$ does $x^{2}+kx+9=0$ have equal roots?
Equal roots require $\Delta=0$: $k^{2}-4(1)(9)=0$, so $k^{2}=36$, giving $k=\pm 6$.
Show the other 30 Equations and Inequalities flashcards
What is meant by an equation 'reducible to a quadratic', and how do you solve one?
An equation that becomes quadratic after a suitable substitution. Substitute $u$ for a repeated expression to get a quadratic in $u$, solve for $u$, then reverse the substitution to find the original variable.
How do you solve a quartic of the form $ax^{4}+bx^{2}+c=0$?
Let $u=x^{2}$ to get $au^{2}+bu+c=0$, solve for $u$, then solve $x^{2}=u$ for each root, taking $x=\pm\sqrt{u}$ (real only where $u \geq 0$).
What substitution turns $x - 5\sqrt{x} + 6 = 0$ into a quadratic, and what is the resulting equation?
Let $u=\sqrt{x}$ (so $x=u^{2}$), giving $u^{2}-5u+6=0$. Solve for $u$, then $x=u^{2}$ (with $u \geq 0$).
What substitution reduces $x^{6}-7x^{3}-8=0$ to a quadratic?
Let $u=x^{3}$, giving $u^{2}-7u-8=0$; solve for $u$, then take the cube root $x=\sqrt[3]{u}$ for each value.
Why must you check solutions after solving an equation involving $\sqrt{x}$ via substitution?
Because $\sqrt{x}\geq 0$, any value of the substitution that is negative must be rejected; squaring/substituting can introduce extraneous solutions that do not satisfy the original equation.
What is the key rule about inequalities when multiplying or dividing by a negative number?
The inequality sign must be reversed (e.g. $-2x < 6$ becomes $x > -3$).
How do you solve a linear inequality such as $3x - 4 \leq 11$?
Treat it like an equation using inverse operations (keeping the sign, but reversing it if multiplying/dividing by a negative): $3x \leq 15$, so $x \leq 5$.
How is the solution of a 'double' inequality like $-3 < 2x+1 \leq 7$ found?
Apply the same operation to all three parts: subtract $1$ to get $-4 < 2x \leq 6$, then divide by $2$ to get $-2 < x \leq 3$.
What is the general process for solving a quadratic inequality?
Rearrange so one side is $0$, find the roots (critical values) by solving the corresponding quadratic equation, sketch the parabola, then read off the $x$-values where the curve is above ($>0$) or below ($<0$) the axis.
For $a>0$, when is $a(x-\alpha)(x-\beta) > 0$ (with $\alpha < \beta$)?
When $x < \alpha$ or $x > \beta$ — i.e. outside the roots, where the upward parabola is above the axis.
For $a>0$, when is $a(x-\alpha)(x-\beta) < 0$ (with $\alpha < \beta$)?
When $\alpha < x < \beta$ — i.e. between the roots, where the upward parabola is below the axis.
Why is sketching the parabola recommended when solving a quadratic inequality?
The sketch shows where the curve lies above or below the $x$-axis relative to the roots, preventing the common error of giving the wrong region (inside vs outside the roots).
What is the solution set of $x^{2} \geq 9$?
$x^{2}-9\geq 0$ gives $(x-3)(x+3)\geq 0$, so $x \leq -3$ or $x \geq 3$.
What is the solution set of $x^{2} < 4$?
$x^{2}-4<0$ gives $(x-2)(x+2)<0$, so $-2 < x < 2$.
What is the special danger when solving an inequality involving an algebraic fraction such as $\frac{1}{x} > 2$?
You cannot simply multiply both sides by $x$, because $x$ may be negative (which would flip the sign). The sign of the denominator is unknown.
What is a reliable technique for solving an inequality with an algebraic fraction (avoiding sign issues)?
Multiply both sides by the square of the denominator (which is always positive, e.g. multiply $\frac{1}{x}>2$ by $x^{2}>0$), then solve the resulting polynomial inequality; alternatively, move everything to one side, combine into a single fraction, and analyse the sign of numerator and denominator using critical values.
When solving $\frac{p(x)}{q(x)} > 0$ by a sign analysis, where are the critical values?
At the zeros of the numerator $p(x)$ and the zeros of the denominator $q(x)$; the latter must be excluded from the solution since the fraction is undefined there.
Solve $\frac{1}{x} > 2$ for real $x$.
Multiply by $x^{2}>0$: $x > 2x^{2}$, i.e. $2x^{2}-x<0$, $x(2x-1)<0$, so $0 < x < \frac{1}{2}$.
Define the modulus (absolute value) function $|x|$.
$$|x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$$ It gives the non-negative magnitude (distance from $0$) of $x$.
What does $|x|$ represent geometrically on the number line?
The distance of $x$ from the origin $0$; more generally $|x-a|$ is the distance between $x$ and $a$.
What does the graph of $y=|x|$ look like?
A V-shape with its vertex at the origin, consisting of the line $y=x$ for $x\geq 0$ and the line $y=-x$ for $x<0$; it is symmetric about the $y$-axis and never goes below the $x$-axis.
How do you solve the equation $|f(x)| = a$ for $a \geq 0$?
Solve the two cases $f(x) = a$ and $f(x) = -a$, then check the solutions are valid. (If $a<0$ there are no solutions.)
How do you solve $|x - 3| = 5$?
Set $x-3 = 5$ or $x-3 = -5$, giving $x = 8$ or $x = -2$.
State the rule for solving $|f(x)| < a$ (with $a>0$).
$|f(x)| < a$ is equivalent to $-a < f(x) < a$.
State the rule for solving $|f(x)| > a$ (with $a>0$).
$|f(x)| > a$ is equivalent to $f(x) > a$ or $f(x) < -a$.
Solve the inequality $|2x - 1| \leq 5$.
$-5 \leq 2x-1 \leq 5$, so $-4 \leq 2x \leq 6$, giving $-2 \leq x \leq 3$.
State the key modulus identity relating $|x|$ to a square root and to $x^{2}$.
$|x| = \sqrt{x^{2}}$ and $|x|^{2} = x^{2}$.
State the multiplicative and quotient properties of the modulus.
$|ab| = |a|\,|b|$ and $\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$ (for $b \neq 0$).
State the triangle inequality for the modulus function.
$|a+b| \leq |a| + |b|$, with equality when $a$ and $b$ have the same sign (or one is zero).
A useful method for solving modulus equations/inequalities like $|f(x)|=|g(x)|$ — what is it?
Square both sides to remove the moduli: $|f(x)|=|g(x)| \iff f(x)^{2}=g(x)^{2}$, then solve the resulting polynomial equation. This is valid because both sides are non-negative.
Test of Mathematics for University Admission (TMUA) Equations and Inequalities FAQ
What is in the Test of Mathematics for University Admission (TMUA) Equations and Inequalities syllabus?
Test of Mathematics for University Admission (TMUA) Equations and Inequalities is split into 3 chapters: Linear and Simultaneous Equations, Quadratic Equations and the Discriminant and Inequalities. Together they contain 10 topics and 20 sub-topics.
How many chapters are in Test of Mathematics for University Admission (TMUA) Equations and Inequalities?
3 chapters, containing 10 topics and 20 sub-topics. The largest chapter is Inequalities, with 4 topics.
Are there flashcards for Test of Mathematics for University Admission (TMUA) Equations and Inequalities?
Yes — a 50-card Equations and Inequalities deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.