IB Mathematics AA HL Flashcards- Solving inequalities

Solving \( f(x) > g(x) \) [AHL]

PAPER 1 (MANUAL)
You may be asked to solve an inequality in paper 1, usually up to quadratics, although the syllabus includes cubics too. If you can rearrange to \( ax^2 + bx + c > 0 \), then either you have:

– If \( a < 0 \), \( \alpha \leq x \leq \beta \)
or
– If \( a > 0 \), \( x < \alpha \) & \( x > \beta \)

Where \( \alpha \& \beta \) are the roots of \( f(x) – g(x) = 0 \).

[q]

Find roots:
\[
2(x^2 – 2x – 15) = 0, 2(x – 5)(x + 3) = 0, x = 5 \text{ or } x = -3.
\]

As \( a > 0 \), and as the quadratic needs to be less than 0 this time:
\[
-3 \leq x \leq 5.
\]

[a]
OTHER
If it is not a quadratic, you may be able to do it algebraically or graphically, using knowledge from topic 2.

Example : Solve \( 1 < \ln x \):

\[
1 < \ln x \implies e^1 < x, \text{so} x > e.
\]

[q]
PAPER 2 (GDC)
I’d suggest simply graphing both functions on your GDC and observe which sections have \( f(x) > g(x) \) and use the intersection function:

– TI-nspire:
New document → 2: ADD GRAPHS → Make \( f_1(x) = f(x) \& f_2(x) = g(x) \) → (MENU) 6: ANALYZE GRAPH → 4: INTERSECTION → Select lower & upper bound.

– TI-84+:
Make \( Y_1 = f(x) \& Y_2 = g(x) \) → GRAPH → (2nd) CALC → 5: INTERSECT → Select first and second graphs.

[x] Exit text