HL AA · Functions

Inequalities — rational and modulus (IB Math AA HL)

Q: Why can't you multiply both sides of an inequality by x − 1?

Because x − 1 is negative for some values of x, and multiplying by a negative number reverses the inequality. You would get the wrong answer for those x. Move everything to one side and use a sign diagram, or multiply by (x − 1)², which is never negative.

Q: Where can I practise inequalities questions?

On the Odd, Even, Inverse, Inequalities and Modulus questions page: 38 exam-style HL AA questions with full mark schemes, plus free step-by-step worked examples.

Q: Can I use my calculator for inequalities?

In HL AA, Paper 1 is non-calculator and the other papers allow a GDC, so learn the exact method as well as the calculator one. The Graphs: intersections, zeros, max and min guide gives the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.

Updated October 2026

Quick answer: For a modulus inequality, |f(x)| < k means −k < f(x) < k, and |f(x)| > k means f(x) < −k or f(x) > k. For a rational inequality, move everything to one side, combine into one fraction, find where the top and bottom are zero, and use a sign diagram. Never multiply both sides by an expression that could be negative.

HL AA Paper 1 tests |x-2| < 3 or (x-1)/(x+2) ≥ 0. Sign diagrams are your friend — never multiply through by a variable expression.

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Key ideas

|x − a| < k means the distance from x to a is less than k: a − k < x < a + k.
Rational inequality: write it as (one fraction) ≥ 0 or ≤ 0. The critical values are the zeros of the numerator and of the denominator.
A zero of the denominator is never part of the solution, even with ≥ or ≤.
Sign diagram: test one value in each interval, or use the signs of each factor, and pick the intervals with the right sign.
Graph both sides on the GDC (Paper 2) to check where one is above the other.

Worked example

Solve (a) |2x − 3| < 5 and (b) (x + 3)/(x − 1) ≤ 2.

(a) −5 < 2x − 3 < 5, so −2 < 2x < 8 and −1 < x < 4.
(b) Subtract 2: (x + 3)/(x − 1) − 2 ≤ 0, so (x + 3 − 2(x − 1))/(x − 1) ≤ 0, which is (5 − x)/(x − 1) ≤ 0.
Critical values: x = 5 (numerator zero) and x = 1 (denominator zero).
Sign of (5 − x)/(x − 1): negative for x < 1, positive for 1 < x < 5, negative for x > 5, and zero at x = 5.
So x < 1 or x ≥ 5 (x = 1 is excluded; x = 5 is included because of ≤).

Answer: (a) −1 < x < 4. (b) x < 1 or x ≥ 5.

Exam-style HL AA questions on this topic

Odd, Even, Inverse, Inequalities and Modulus questions — 38 questions with mark schemes, plus free worked examples
Polynomials and Rational Functions questions — 79 questions with mark schemes, plus free worked examples
Transformations and Modulus questions — 20 questions with mark schemes, plus free worked examples
All HL AA Functions questions by topic

Do it on your GDC

A calculator is allowed on AA Paper 2 and Paper 3, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

Graphs: intersections, zeros, max and min on your calculator

See it on a graph

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HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
IB Maths prediction papers — full papers in the IB format, with markschemes

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FAQ

Why can't you multiply both sides of an inequality by x − 1?

Because x − 1 is negative for some values of x, and multiplying by a negative number reverses the inequality. You would get the wrong answer for those x. Move everything to one side and use a sign diagram, or multiply by (x − 1)², which is never negative.

Where can I practise inequalities questions?

On the Odd, Even, Inverse, Inequalities and Modulus questions page: 38 exam-style HL AA questions with full mark schemes, plus free step-by-step worked examples.

Can I use my calculator for inequalities?

In HL AA, Paper 1 is non-calculator and the other papers allow a GDC, so learn the exact method as well as the calculator one. The Graphs: intersections, zeros, max and min guide gives the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.

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