IB Maths AA HL Topic 2 — Functions Paper 2 HL only ~7 min read

Solving Inequalities Graphically

When an inequality mixes function types — exponential vs polynomial, log vs linear, trig vs anything — algebra usually can’t isolate x. The fix is the same as for equations: rearrange to one side equals zero, sketch on the GDC, and read off the regions where the curve sits above or below the x-axis. The only watch-out is that multiplying or dividing by negatives flips the inequality sign — and you should never multiply by anything containing the variable.

📘 What you need to know

Standard set-up: rewrite f(x) ≤ g(x) as f(x) − g(x) ≤ 0, then sketch y = f(x) − g(x).
Solutions are intervals where the curve is below (f − g ≤ 0) or above (f − g ≥ 0) the x-axis.
Inclusive vs strict: ≤ and ≥ include the x-intercepts; < and > exclude them.
Flip the sign when multiplying or dividing both sides by a negative number.
Never multiply by a variable expression whose sign is unknown — it might be positive or negative depending on x, and the inequality direction would change differently in each case.
Safe rearrangement: only add and subtract terms across the inequality. Adding/subtracting never changes the direction.
Round to 3 sf when reading values from a GDC, unless the question specifies otherwise.
State the domain if the original functions have restrictions (e.g. ln x requires x > 0).

The general graphical method

Standard set-up f(x) ≤ g(x) ⟺ f(x) − g(x) ≤ 0

One side becomes zero, the other becomes a single function you can plot. The roots of f − g are the x-values where the two original curves cross — and between those crossings, one of f, g is bigger than the other.

f(x) ≤ g(x)

curve below or on x-axis

read off intervals where f − g ≤ 0

f(x) ≥ g(x)

curve above or on x-axis

read off intervals where f − g ≥ 0

Reading off solution intervals from the difference graph

When to flip the inequality sign

Sign-flip rule multiply or divide both sides by a negative number → flip the inequality

Always safe

add or subtract
multiply by x², |x|, e^x

these are always non-negative or strictly positive

Risky

multiply or divide by x
multiply by any expression with unknown sign

can flip the inequality differently depending on x

Safest move: never multiply or divide an inequality by a variable. Just rearrange by adding and subtracting terms — that always preserves the direction. The graphical method then handles the rest.

The GDC recipe

🧭 Recipe — solving an inequality graphically

Rearrange the inequality to f(x) − g(x) ≤ 0 (or ≥ 0). Use only addition and subtraction.
Plot y = f(x) − g(x) on the GDC. Use a wide window so all roots are visible.
Find the roots using the GDC’s zero/root function. Round to 3 sf.
Read off the regions where the curve sits below (≤ 0) or above (≥ 0) the x-axis.
Match the inequality type: ≤/≥ uses inclusive endpoints; </> uses strict.
State the solution as one or more intervals. If multiple, separate with “or”.

If the question gives a domain (e.g. 0 ≤ x ≤ π), restrict your answer to that range. Otherwise, the full real line — including any natural domain restrictions of the functions involved (like x > 0 for ln x).

Worked examples

WE 1

Solve an exponential vs polynomial inequality

Solve e^x ≤ x² + 2, giving your answer to 3 sf.

Step 1: Rearrange so one side is zero e^x − x² − 2 ≤ 0 Step 2: Sketch y = e^x − x² − 2 on the GDC as x → −∞, x² dominates → y → −∞ as x → +∞, e^x dominates → y → +∞ Step 3: Find the root using GDC single zero at x ≈ 1.32 Step 4: Curve below x-axis for x ≤ 1.32 x ≤ 1.32 (3 sf) no algebraic isolation possible — the GDC is the only practical route

WE 2

Solve a logarithmic inequality

Solve ln(x) ≥ x − 2, giving your answer to 3 sf.

Step 1: Rearrange — note domain x > 0 for ln ln(x) − x + 2 ≥ 0 Step 2: Sketch y = ln(x) − x + 2 on x > 0 as x → 0⁺, ln(x) → −∞ → y → −∞ as x → +∞, −x dominates → y → −∞ curve has a maximum, two zeros Step 3: Find roots on GDC zeros at x ≈ 0.159 and x ≈ 3.15 Step 4: Curve above x-axis between the roots 0.159 ≤ x ≤ 3.15 (3 sf) don’t forget to state the domain — ln(x) is undefined for x ≤ 0

WE 3

Solve a trigonometric inequality on a given domain

Solve sin(x) > x3 for 0 ≤ x ≤ π, giving your answer to 3 sf.

Step 1: Set GDC to radian mode and rearrange sin(x) − x/3 > 0 Step 2: Sketch y = sin(x) − x/3 on [0, π] at x = 0: sin(0) − 0 = 0 (on the axis) curve rises above x-axis, then comes back down Step 3: Find the second root on GDC first root at x = 0; second root at x ≈ 2.28 Step 4: Strict inequality — exclude both endpoints where y = 0 0 < x < 2.28 (3 sf) always check the GDC’s angle mode — degrees vs radians silently destroys trig answers

WE 4

Polynomial inequality with messy roots

Solve x⁴ − 5x² + 2 < 0, giving your answer to 3 sf.

Step 1: Already in f(x) < 0 form Step 2: Sketch y = x⁴ − 5x² + 2 on the GDC positive leading, even degree → both ends to +∞ curve dips below x-axis twice (symmetric quartic) Step 3: Find the four roots x ≈ ±0.662 and x ≈ ±2.14 Step 4: Curve below x-axis on two intervals −2.14 < x < −0.662 or 0.662 < x < 2.14 (3 sf) factorising via u = x² works in principle, but the GDC is much faster on a calculator paper

WE 5

Cubic inequality with multiple regions

Solve x³ + 1 ≥ 5x, giving your answer to 3 sf.

Step 1: Rearrange x³ − 5x + 1 ≥ 0 Step 2: Sketch y = x³ − 5x + 1 on the GDC positive leading, odd degree → −∞ on left, +∞ on right three real roots — graph weaves around x-axis Step 3: Find roots x ≈ −2.33, x ≈ 0.201, x ≈ 2.13 Step 4: Curve above x-axis on two intervals −2.33 ≤ x ≤ 0.201 or x ≥ 2.13 (3 sf) cubic with three real roots → two solution intervals; cubics with one real root → just one interval

WE 6

Find the number of integer solutions

Use a graphical method to find the number of integer values of x satisfying e^−x²/4 > 0.6.

Step 1: Rearrange e^(−x²/4) − 0.6 > 0 Step 2: Sketch y = e^(−x²/4) − 0.6 bell-shaped curve, max at x = 0 (value 0.4) crosses x-axis at two symmetric points Step 3: Find roots x ≈ ±1.43 Step 4: Solution interval and integer count −1.43 < x < 1.43 integers in this interval: −1, 0, 1 3 integer solutions “how many” questions just need a count — no need to state the integers themselves unless asked

💡 Top tips

Rearrange to one side first. A single curve crossing zero is much easier to read than two curves crossing each other.
Use a wide GDC window initially, then zoom in on each root. Missing a root off-screen costs marks.
Match the inequality type: ≤ and ≥ include the boundary points (x-intercepts); < and > exclude them.
Use only addition and subtraction when rearranging. Multiplying both sides by a variable is the most common source of wrong answers.
Always check the GDC’s angle mode for trig inequalities. Radians vs degrees flips every value.
State domain restrictions: ln x requires x > 0; √x requires x ≥ 0. The solution must be inside the domain.
Sketch the graph in your working — examiners look for evidence of the method, not just the final interval.

⚠ Common mistakes

Multiplying by x without considering its sign. The inequality direction can flip depending on whether x is positive or negative — leading to wrong answers in part of the domain.
Forgetting to flip the sign when multiplying or dividing by a negative number.
Reading off y-coordinates instead of x-values for the boundaries. Solutions are x-values.
Mixing inclusive and strict endpoints. ≤ in the question means ≤ in the answer.
Missing roots off-screen. Always confirm with a wide window and check end behaviour.
Ignoring the natural domain of the functions. ln x > 1 only makes sense for x > 0.
Wrong angle mode on the GDC for trig — silent killer of trig inequality questions.

When the inequality is purely polynomial — and especially when you need an algebraic answer — there’s a faster method that doesn’t need a GDC. The next note, Polynomial Inequalities, covers the sign-table approach: factor the polynomial, work out the sign of each factor in each interval, and read off the solution. Quicker than graphical for clean factorisations.

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