What is the difference between an equation and an inequality in IB DP Mathematics?

In IB DP Mathematics, an equation is a mathematical statement that asserts the equality of two expressions, typically involving variables. For example, 2x + 3 = 7 is an equation. An inequality, on the other hand, is a relation that holds between two values when they are different, often using symbols like , ≤, or ≥. For instance, 2x + 3 > 7 is an inequality. Equations are solved to find exact values, while inequalities are solved to find a range of values.

How do you solve linear equations in one variable?

To solve a linear equation in one variable, such as ax + b = c, you need to isolate the variable on one side of the equation. This involves performing inverse operations to both sides of the equation. Start by subtracting or adding terms to both sides to eliminate constants, then divide or multiply to solve for the variable. For example, to solve 2x + 3 = 7, subtract 3 from both sides to get 2x = 4, then divide by 2 to find x = 2.

What strategies can be used to solve quadratic inequalities?

Solving quadratic inequalities involves finding the values of the variable that make the inequality true. Start by solving the corresponding quadratic equation to find the roots. Use these roots to divide the number line into intervals. Test each interval to determine where the inequality holds true. Graphing the quadratic function can also help visualize the solution. For example, for x^2 - 4 < 0, solve x^2 - 4 = 0 to find roots x = -2 and x = 2, then test intervals to find the solution is -2 < x < 2.

How can systems of equations be solved using the substitution method?

The substitution method involves solving one of the equations in a system for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable. Solve this equation to find the value of one variable, then substitute back to find the other. For example, for the system y = 2x + 3 and x + y = 7, substitute y = 2x + 3 into the second equation to get x + (2x + 3) = 7, solve for x, and then use this value to find y.

What are some common mistakes to avoid when solving inequalities?

When solving inequalities, a common mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. Additionally, students often overlook testing intervals when dealing with quadratic inequalities. It's also important to ensure that all possible solutions are considered, especially when dealing with compound inequalities. Always check your solutions by substituting them back into the original inequality to verify their correctness.

Why is "Cross-multiplying a rational inequality by the denominator without checking its sign." a common mistake in Equations and Inequalities?

Why it happens: Habit from equations. How to avoid it: Bring everything to one side and use critical-point sign analysis. Cross-multiplying by an algebraic expression that could be negative reverses the inequality.

Why is "Treating $|f| = g$ as just $f = \pm g$ without restricting $g \ge 0$." a common mistake in Equations and Inequalities?

Why it happens: Forgetting that the RHS must be non-negative. How to avoid it: If g < 0, the equation has no solutions. Otherwise solve f = g AND f = -g, then verify each in the original.

Why is "Squaring an inequality introduces extraneous solutions." a common mistake in Equations and Inequalities?

Why it happens: Squaring is not order-preserving for negatives. How to avoid it: Always check boundary solutions in the original after squaring. Reject any that fail.

Equations and InequalitiesIB Maths AA HL

Equations and Inequalities

Q: How can systems of equations be solved using the substitution method?

The substitution method involves solving one of the equations in a system for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable. Solve this equation to find the value of one variable, then substitute back to find the other. For example, for the system y = 2x + 3 and x + y = 7, substitute y = 2x + 3 into the second equation to get x + (2x + 3) = 7, solve for x, and then use this value to find y.

Q: What are some common mistakes to avoid when solving inequalities?

When solving inequalities, a common mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. Additionally, students often overlook testing intervals when dealing with quadratic inequalities. It's also important to ensure that all possible solutions are considered, especially when dealing with compound inequalities. Always check your solutions by substituting them back into the original inequality to verify their correctness.

Q: Why is "Cross-multiplying a rational inequality by the denominator without checking its sign." a common mistake in Equations and Inequalities?

Why it happens: Habit from equations. How to avoid it: Bring everything to one side and use critical-point sign analysis. Cross-multiplying by an algebraic expression that could be negative reverses the inequality.

Q: Why is "Treating $|f| = g$ as just $f = \pm g$ without restricting $g \ge 0$." a common mistake in Equations and Inequalities?

Why it happens: Forgetting that the RHS must be non-negative. How to avoid it: If g < 0, the equation has no solutions. Otherwise solve f = g AND f = -g, then verify each in the original.

Q: Why is "Squaring an inequality introduces extraneous solutions." a common mistake in Equations and Inequalities?

Why it happens: Squaring is not order-preserving for negatives. How to avoid it: Always check boundary solutions in the original after squaring. Reject any that fail.

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Equations and Inequalities — IB Maths AA HL: linear systems, polynomial, rational, modulus and GDC graphical methods

AA HL extends SL with rational inequalities, modulus (absolute value) inequalities and 3×3 linear systems. This note covers exact-form solving, the sign-flip rule, modulus manipulation, and GDC graphical methods for Paper 2.

What you’ll learn

Mapped to the IB DP Maths AA HL subject guide (2021 onwards (applies to 2026 exams)).

AO1 — Solve linear, quadratic and polynomial equations exactly.
AO1 — Apply the sign-flip rule for inequalities.
AO2 — Solve rational and modulus inequalities by critical-point analysis.
AO2 — Use GDC for transcendental equations (Paper 2).
AO2 — Solve 3×3 linear systems.
AO3 — Justify extraneous-root rejection and domain restrictions.

Linear systems (2 and 3 unknowns)

Substitution, elimination, GDC matrix solver.

2×2 systems. Use elimination or substitution.

3×3 systems. On Paper 1 (no calc), use elimination/substitution. On Paper 2, the GDC's matrix solver $A^{-1} \mathbf{b}$ is fastest — but you must show the matrix set-up.

Worked example (3×3). Solve: $x + y - z = 4, \quad 2x - y + z = 1, \quad x + 2y - z = 7.$

From first two: add → $3x = 5 \Rightarrow x = 5/3$ . From first: $y - z = 4 - 5/3 = 7/3$ .

From first and third: subtract → $-y = -3 \Rightarrow y = 3$ . Then $z = y - 7/3 = 2/3$ .

Check in second: $2(5/3) - 3 + 2/3 = 10/3 - 3 + 2/3 = 12/3 - 3 = 1$ ✓.

Three-variable systems with no unique solution:

Inconsistent (no solution): equations contradict.
Dependent (infinite solutions): equations are linearly dependent.

Both occur in HL Paper 3 problems where parameters are varied.

2×2: substitution or elimination.
3×3: elimination chains; GDC matrix on Paper 2.
Check by substituting into the original equations.

Rational and modulus inequalities (HL only)

Critical points + sign analysis.

Rational inequality. $\dfrac{P(x)}{Q(x)} > 0$ or other inequalities.

Method.

Bring everything to one side: $\dfrac{P}{Q} - c > 0$ etc.
Find critical points: where numerator = 0 (roots) or denominator = 0 (excluded).
Sign chart: test the sign of $P/Q$ between consecutive critical points.
Read off the region.

Worked example. Solve $\dfrac{x - 3}{x + 2} \le 0$ .

Critical points: $x = 3$ (num = 0) and $x = -2$ (denom = 0, EXCLUDED).

Sign chart:

Interval	$x - 3$	$x + 2$	$\dfrac{x-3}{x+2}$
$x < -2$	$-$	$-$	$+$
$-2 < x < 3$	$-$	$+$	$-$
$x > 3$	$+$	$+$	$+$

We want $\le 0$ : $-2 < x \le 3$ . (Open at $-2$ because excluded; closed at 3 because numerator = 0 satisfies $\le 0$ .)

Modulus inequalities.

$|x| < a$ ⇔ $-a < x < a$ (for $a > 0$ ).
$|x| > a$ ⇔ $x < -a$ or $x > a$ .
$|f(x)| < g(x)$ ⇔ $-g(x) < f(x) < g(x)$ when $g(x) > 0$ .

Worked example. Solve $|2x - 5| \le 7$ .

$-7 \le 2x - 5 \le 7 \Rightarrow -2 \le 2x \le 12 \Rightarrow -1 \le x \le 6$ .

Worked example (graphical). Solve $|x - 1| > 2|x|$ .

Square both sides (preserves inequality since both sides $\ge 0$ , taking $|x| \ne 0$ ): $(x - 1)^2 > 4x^2 \Rightarrow x^2 - 2x + 1 > 4x^2 \Rightarrow -3x^2 - 2x + 1 > 0 \Rightarrow 3x^2 + 2x - 1 < 0$ .

Factor: $(3x - 1)(x + 1) < 0$ . Solution: $-1 < x < 1/3$ .

(Check $x = 0$ : $|0 - 1| = 1 > 0 = 2|0|$ ✓ — so $x = 0$ included.)

(Sometimes squaring introduces extraneous solutions — verify boundary points.)

Rational inequality: don't cross-multiply by $Q(x)$ without sign awareness.
Sign chart with critical points is the robust method.
Modulus: $|f| < g$ ⇔ $-g < f < g$ .
Squaring works when both sides ≥ 0 — verify after.

GDC graphical methods + hidden quadratics

Paper 2 staple.

Hidden quadratics. Substitution turns transcendental equations into quadratics.

Worked example. Solve $e^{2x} - 7e^x + 12 = 0$ .

Let $u = e^x$ (so $u > 0$ ): $u^2 - 7u + 12 = 0 \Rightarrow (u - 3)(u - 4) = 0$ .

$u = 3$ or $u = 4$ — both positive, both valid.

$x = \ln 3$ or $x = \ln 4$ .

Worked example. Solve $2\sin^2 x - 3\sin x + 1 = 0$ for $0 \le x \le 2\pi$ .

Let $u = \sin x$ : $2u^2 - 3u + 1 = 0 \Rightarrow (2u - 1)(u - 1) = 0$ .

$u = 1/2$ or $u = 1$ .

$\sin x = 1/2$ : $x = \pi/6, 5\pi/6$ . $\sin x = 1$ : $x = \pi/2$ .

Three solutions.

GDC graphical for transcendental. Plot $y_1 =$ LHS and $y_2 =$ RHS; use intersect. State the equation and the window. Report to required s.f.

Worked example. Solve $e^x = 4 - x^2$ to 3 s.f. (Paper 2).

GDC: plot both, intersect. Approximately $x \approx -1.96$ and $x \approx 1.06$ .

Always check both solutions visually — if you only spotted one intersection, you missed one.

Hidden quadratic: substitute $u = e^x$ , $u = \sin x$ , $u = \sqrt{x}$ , etc.
Validate the $u$ values (positive for $e^x$ , in $[-1, 1]$ for $\sin$ ).
GDC graphical method: state the equation + window.

How it’s examined

Paper 1: exact algebra, sign-flip rule, rational and modulus inequalities. Paper 2: GDC for transcendental and 3×3 systems. Paper 3: extended problems combining multiple inequality types. Examiner reports flag students who cross-multiply by an algebraic expression without checking its sign.

Sources: IB Diploma Programme Mathematics: Analysis and Approaches subject guide (IBO, official). Last reviewed 2026-05-31.

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Step-by-step worked examples — Equations and Inequalities

Step-by-step solutions to past-paper-style questions on equations and inequalities, written exactly the way a tutor would explain them at the board.

1Solve a 3×3 system
Building confidence• system
Question
Solve $x + y + z = 6$ , $2x - y + z = 3$ , $-x + y + 2z = 7$ .
Step-by-step solution
1. Step 1
  Add first and third: $2y + 3z = 13$ .
2. Step 2
  From first and second (subtract): $-x + 2y = 3 \Rightarrow x = 2y - 3$ .
3. Step 3
  Substitute in first: $(2y - 3) + y + z = 6 \Rightarrow z = 9 - 3y$ .
4. Step 4
  From $2y + 3z = 13$ : $2y + 3(9 - 3y) = 13 \Rightarrow -7y = -14 \Rightarrow y = 2$ .
5. Step 5
  Then $x = 2(2) - 3 = 1$ and $z = 9 - 6 = 3$ .
Answer
$(x, y, z) = (1, 2, 3)$ .
2Rational inequality
Stretch• inequality
Question
Solve $\dfrac{x + 1}{x - 2} \ge 0$ .
Step-by-step solution
1. Step 1
  Critical points: $x = -1$ (num = 0) and $x = 2$ (denom = 0, excluded).
2. Step 2
  Test regions. $x < -1$ : $(-)/(-) = +$ . $-1 < x < 2$ : $(+)/(-) = -$ . $x > 2$ : $(+)/(+) = +$ .
3. Step 3
  $\ge 0$ where positive OR num = 0: $x \le -1$ OR $x > 2$ .
Answer
$x \le -1$ or $x > 2$ .
3Modulus inequality
Building confidence• modulus
Question
Solve $|3x + 4| \le 8$ .
Step-by-step solution
1. Step 1
  $-8 \le 3x + 4 \le 8$ .
2. Step 2
  Subtract 4: $-12 \le 3x \le 4$ .
3. Step 3
  Divide by 3: $-4 \le x \le 4/3$ .
Answer
$-4 \le x \le 4/3$ .
4Hidden quadratic — trig
Stretch• substitution
Question
Solve $2\cos^2 x + \cos x - 1 = 0$ for $0 \le x \le 2\pi$ .
Step-by-step solution
1. Step 1
  Let $u = \cos x$ : $2u^2 + u - 1 = 0 \Rightarrow (2u - 1)(u + 1) = 0$ .
2. Step 2
  $u = 1/2$ : $\cos x = 1/2 \Rightarrow x = \pi/3, 5\pi/3$ .
3. Step 3
  $u = -1$ : $\cos x = -1 \Rightarrow x = \pi$ .
Answer
$x = \pi/3, \pi, 5\pi/3$ .

Key Definitions and Keywords — Equations and Inequalities

Definitions to memorise and the exact keywords mark schemes credit for equations and inequalities answers — sharpened from recent examiner reports for the 2026 IB DP Maths AA HL sitting.

Modulus / absolute value
Examiner keyword
$|x| = x$ if $x \ge 0$ , $-x$ if $x < 0$ . Geometric: distance from 0.
Critical point (rational inequality)
Examiner keyword
$x$ values where numerator or denominator equals zero. Sign chart is constructed between consecutive critical points.

Common Mistakes and Misconceptions — Equations and Inequalities

The traps other students keep falling into on equations and inequalities questions — taken from recent IB DP Maths AA HL examiner reports and mark schemes — and how to avoid them.

✕Cross-multiplying a rational inequality by the denominator without checking its sign.
Why it happens
Habit from equations.
How to avoid it
Bring everything to one side and use critical-point sign analysis. Cross-multiplying by an algebraic expression that could be negative reverses the inequality.
✕Treating $|f| = g$ as just $f = \pm g$ without restricting $g \ge 0$ .
Why it happens
Forgetting that the RHS must be non-negative.
How to avoid it
If $g < 0$ , the equation has no solutions. Otherwise solve $f = g$ AND $f = -g$ , then verify each in the original.
✕Squaring an inequality introduces extraneous solutions.
Why it happens
Squaring is not order-preserving for negatives.
How to avoid it
Always check boundary solutions in the original after squaring. Reject any that fail.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Equations and Inequalities — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Equations and Inequalities

Short Study Notes in the form of Slides

Detailed Notes

What you’ll learn

How it’s examined

1Solve a 3×3 system

2Rational inequality

3Modulus inequality

4Hidden quadratic — trig

Modulus / absolute value

Critical point (rational inequality)

✕Cross-multiplying a rational inequality by the denominator without checking its sign.

✕Treating ∣f∣=g|f| = g∣f∣=g as just f=±gf = \pm gf=±g without restricting g≥0g \ge 0g≥0.

✕Squaring an inequality introduces extraneous solutions.

Practice questions

Past paper style quiz

Equations and Inequalities — frequently asked questions

✕Treating $|f| = g$ as just $f = \pm g$ without restricting $g \ge 0$ .