Solving Quadratic Inequalities Flashcards

All 5 cards in this deck

What are the steps to solve a quadratic inequality?

e.g. $x^2 - x - 6 < 0$

1. Write it as $ax^2+bx+c$ compared with $0$: already done
2. Factorise to get the critical values: $(x-3)(x+2)$, so $x = 3$ and $x = -2$
3. Choose the region using a sketch: $-2 < x < 3$

For $a < b$, what is the solution set of $(x-a)(x-b) < 0$?

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

$a < x < b$ (the values between the critical values).
e.g. $-5 < x < 2$

For $a < b$, what is the solution set of $(x-a)(x-b) > 0$?

e.g. $(x+2)(x-\tfrac{1}{2}) > 0$

$x < a$ or $x > b$ (the values outside the critical values).
e.g. $x < -2$ or $x > \frac{1}{2}$

True or false? The solution of $x^2 > 9$ is $x > 3$.

False. The solution is $x < -3$ or $x > 3$.

What are the steps to find the set of possible values of $x$ when one area must be greater than another?

e.g. rectangle area $(x-1)(3x-2)$ greater than triangle area $\tfrac{1}{2}(2x \times x)$

1. Write the inequality: $(x-1)(3x-2) > x^2$
2. Expand and rearrange to $ax^2+bx+c>0$: $2x^2-5x+2>0$
3. Solve and reject values making a length negative: critical values $\frac{1}{2}$ and $2$, so $x > 2$