Test of Mathematics for University Admission (TMUA) Equations and Inequalities Flashcards

24 sample cards from the Equations and Inequalities deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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 $x \leq 5$.

2. 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 $-2 < x \leq 3$.

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.

4. 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.

5. 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.

6. 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).

7. 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 $x \geq 3$.

8. What is the solution set of $x^{2} < 4$?

   $x^{2}-4<0$ gives $(x-2)(x+2)<0$, so $-2 < x < 2$.

9. 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.

10. 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.

11. 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.

12. 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 $0 < x < \frac{1}{2}$.

13. 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 $x$.

14. 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 $a$.

15. 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.

16. 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.)

17. How do you solve $|x - 3| = 5$?

    Set $x-3 = 5$ or $x-3 = -5$, giving $x = 8$ or $x = -2$.

18. State the rule for solving $|f(x)| < a$ (with $a>0$).

    $|f(x)| < a$ is equivalent to $-a < f(x) < a$.

19. State the rule for solving $|f(x)| > a$ (with $a>0$).

    $|f(x)| > a$ is equivalent to $f(x) > a$ or $f(x) < -a$.

20. Solve the inequality $|2x - 1| \leq 5$.

    $-5 \leq 2x-1 \leq 5$, so $-4 \leq 2x \leq 6$, giving $-2 \leq x \leq 3$.

21. State the key modulus identity relating $|x|$ to a square root and to $x^{2}$.

    $|x| = \sqrt{x^{2}}$ and $|x|^{2} = x^{2}$.

22. State the multiplicative and quotient properties of the modulus.

    $|ab| = |a|\,|b|$ and $\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$ (for $b \neq 0$).

23. 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).

24. 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.

What this deck covers

Equations and Inequalities flashcards FAQ