Solving inequalities: when does the sign flip? Linear, compound "and"/"or" and quadratic inequalities, with examples

Q: How do "and" and "or" differ in compound inequalities?

"And" means both conditions hold at once, so take the overlap on the number line; "or" means either is enough, so take both pieces together. If the two pieces of an "and" inequality do not overlap, there is no solution.

Q: How do I tell whether a quadratic inequality's answer is between the roots or outside them?

First make the x² coefficient positive. The parabola then opens upwards, so " 0" or "≥ 0" is outside them. If unsure, test one number from each interval.

Sprout Editorial Team・30 October 2026・Updated 31 October・4 min read

Inequalities follow the same rules as equations with one addition: when you multiply or divide both sides by a negative number, the inequality sign flips; adding or subtracting any number, or multiplying or dividing by a positive number, leaves it unchanged. Junior secondary students learn linear inequalities in one unknown; the senior Compulsory Part adds compound inequalities joined by the logical connectives "and" or "or", quadratic inequalities and linear programming. "And" takes the overlap of two solutions; "or" takes everything either one covers. For a quadratic inequality, move every term to one side, find the two roots, then see whether the parabola is above or below the x-axis: with a positive x² coefficient, "> 0" lies outside the roots and "< 0" between them. Solving quadratic inequalities algebraically and linear programming are non-foundation topics, which appear only in Section B of the HKDSE papers. Substituting a number from your answer back into the question checks whether the direction is right.

A Hong Kong student's hands tidy scattered wooden blocks and a pencil into a neat row on a wooden desk, half messy, half tidy.

Key points

Multiplying or dividing both sides by a negative number flips the sign; adding, subtracting, or multiplying or dividing by a positive number does not.
Moving the x terms to the side where their coefficient is positive avoids dividing by a negative number.
"And" takes the overlap of two solutions; "or" takes everything either one covers.
For quadratic inequalities, find the two roots; with a positive x² coefficient, "< 0" is between them and "> 0" outside them.
The algebraic method for quadratic inequalities and linear programming are non-foundation topics, only in HKDSE Section B.
Test one number inside and one outside your answer to check the direction.

6 hoursSuggested teaching time, junior linear inequalitiesSource: 初中數學科學習內容（數學教育學習領域課程指引補充文件，2017）
16 hoursSuggested teaching time, senior inequalities and linear programmingSource: 高中數學科學習內容（數學教育學習領域課程指引補充文件，2017）
Section B onlyWhere non-foundation topics appear in the HKDSESource: 2027年香港中學文憑考試數學評核大綱

Where inequalities sit in the secondary curriculum

Inequalities are learnt in two stages. The junior secondary unit "Linear inequalities in one unknown" has a suggested teaching time of 6 hours. It covers writing word statements as inequalities, showing x > a, x ≥ a, x < a and x ≤ a on a number line, the basic properties of inequalities, and solving linear inequalities in one unknown and related problems.

The senior Compulsory Part unit "Inequalities and linear programming" has a suggested teaching time of 16 hours. It covers [compound linear inequalities involving the logical connectives "and" or "or", the triangle inequality, solving quadratic inequalities by the graphical and algebraic methods, the graphs of linear inequalities in two unknowns, and linear programming problems](s2). Solving quadratic inequalities algebraically, linear inequalities in two unknowns and linear programming are non-foundation topics. In Sprout's course, linear inequalities in one unknown are in Secondary 3 and this senior unit is in Secondary 5.

The difference matters in the HKDSE. Section A of Paper 1 and Paper 2 covers foundation topics only, while Section B covers both foundation and non-foundation topics. So compound inequalities and the graphical method for quadratic inequalities can appear in Section A, while the algebraic method and linear programming appear only in Section B.

Curriculum content related to inequalities (EDB learning content, 2017)
Stage	Learning objectives	Foundation / non-foundation
Junior secondary	Linear inequalities in one unknown: word statements, number lines, basic properties, solving, problems	Foundation
Senior Compulsory Part	Compound linear inequalities ("and", "or"); the triangle inequality	Foundation
Senior Compulsory Part	Solving quadratic inequalities by the graphical method	Foundation
Senior Compulsory Part	Solving quadratic inequalities by the algebraic method	Non-foundation
Senior Compulsory Part	Graphs of linear inequalities in two unknowns, systems of them, linear programming problems	Non-foundation

Basic properties: why does multiplying or dividing by a negative number flip the sign?

The basic properties listed in the curriculum include: if a > b and b > c, then a > c; if a > b, then a ± c > b ± c; and if a > b and c is positive, then ac > bc, but if c is negative, then ac < bc, with the same holding for division. In short, adding or subtracting any number and multiplying or dividing by a positive number keep the direction; only multiplying or dividing by a negative number flips it.

A number line shows why. 3 > 2, but after multiplying both by −1, −3 lies to the left of −2, so −3 < −2. Multiplying by a negative number mirrors the whole number line, which reverses the order of numbers, so the sign must flip. For example, dividing −2x > 6 by −2 gives x < −3; writing x > −3 means the flip was missed.

A safer habit is to move the x terms to the side where their coefficient is positive, so you never divide by a negative number. For 5 − 3x ≥ 2x − 10, add 3x and 10 to both sides to get 15 ≥ 5x, so x ≤ 3, without any division by a negative.

Solving a linear inequality

Solving linear inequalities and word problems

The steps are the same as for a linear equation: expand brackets, move terms, collect like terms, and finally divide by the coefficient of x, watching whether that coefficient is positive or negative. For 3(x − 1) ≤ 5x + 7: expanding gives 3x − 3 ≤ 5x + 7; subtracting 3x and 7 from both sides gives −10 ≤ 2x; dividing by the positive number 2 gives x ≥ −5.

With fractions, first multiply both sides by the lowest common multiple of the denominators. For (x − 1)/2 − (x + 2)/3 > 1, multiplying by 6 gives 3(x − 1) − 2(x + 2) > 6, that is 3x − 3 − 2x − 4 > 6, so x > 13. Multiplying by the positive number 6 keeps the direction.

Word problems turn phrases such as "at least", "at most", "no more than" and "no fewer than" into inequalities, and the curriculum asks students to express word statements as inequalities. Example: notebooks cost $12 each, you have already bought a $25 pencil case, and you can spend at most $100 in all. How many notebooks can you buy? With n notebooks, 12n + 25 ≤ 100, so 12n ≤ 75 and n ≤ 6.25. Since n is a whole number, the most is 6. For "greatest integer" questions like this, read the answer from the number line rather than rounding 6.25.

On a number line, > or < uses a hollow dot, meaning the end point is not included; ≥ or ≤ uses a solid dot, meaning it is included.

Secondary 3 maths: linear inequalities in one unknown

Compound inequalities: the logical connectives "and" and "or"

The senior curriculum asks students to solve compound inequalities involving the logical connectives "and" or "or". Solve each part separately, then draw both solutions on one number line: "and" means both conditions hold at once, so take only the overlap; "or" means either one is enough, so take everything either part covers.

Example 1 ("and"): 2x − 1 > 3 and x + 4 ≤ 10. Solving each gives x > 2 and x ≤ 6, which overlap in 2 < x ≤ 6. Example 2 ("or"): 3 − x > 5 or 2x ≥ 8. Solving each gives x < −2 or x ≥ 4; the answer is these two pieces, and the part in between, −2 ≤ x < 4, is not included. Example 3: x > 5 and x < 1 do not overlap on the number line, so there is no solution.

The triangle inequality belongs to the same learning objective: any two sides of a triangle add up to more than the third side. With sides 4, 7 and x, we need 4 + 7 > x, 4 + x > 7 and 7 + x > 4, which together give 3 < x < 11: the unknown side lies between the difference and the sum of the other two.

Quadratic inequalities: find the two roots, then look at the parabola

The curriculum lists solving quadratic inequalities by the graphical method and by the algebraic method separately; the first is a foundation topic and the second non-foundation. Both start the same way: move every term to one side so the other side is 0, then factorise or use the quadratic formula to find the two roots.

Graphical method: sketch y = ax² + bx + c through its two roots. When a > 0 the parabola opens upwards, so "> 0" is where the curve is above the x-axis, outside the roots, and "< 0" is where it is below, between the roots. For example, x² − 5x + 6 > 0 is (x − 2)(x − 3) > 0, so x < 2 or x > 3; with < 0 instead, the answer is 2 < x < 3.

Algebraic method: list the sign of each factor on each interval. For (x + 2)(x − 3): when x < −2 both factors are negative, so the product is positive; when −2 < x < 3 the product is negative; when x > 3 it is positive. So (x + 2)(x − 3) ≤ 0 gives −2 ≤ x ≤ 3.

Watch three cases. First, if the x² coefficient is negative, multiply both sides by −1 and flip the sign: −x² + x + 6 ≥ 0 becomes x² − x − 6 ≤ 0, so −2 ≤ x ≤ 3. Second, if it does not factorise, use the quadratic formula: x² − 2x − 1 < 0 has roots 1 ± √2, so 1 − √2 < x < 1 + √2. Third, if the discriminant is negative, the curve does not meet the x-axis: x² + 2x + 5 has discriminant 4 − 20 = −16 and opens upwards, so x² + 2x + 5 > 0 for every real x, and x² + 2x + 5 < 0 has no real solution.

The senior curriculum also asks students to use the graph of y = f(x) to solve inequalities such as f(x) > k and f(x) < k. The idea is the same: see where the curve is above or below the line y = k.

Key formulas at a glance

Number and algebraInequalities−2x > 6 ⇒ x < −3Multiplying or dividing by a negative number flips the sign
Number and algebraQuadratic formulax = −b ± √b² − 4ac2aSolves ax² + bx + c = 0
Number and algebraDiscriminantΔ = b² − 4acΔ > 0: two unequal real roots; Δ = 0: two equal real roots; Δ < 0: no real roots

Linear inequalities in two unknowns and linear programming (non-foundation)

Students taking the non-foundation topics also learn to represent linear inequalities in two unknowns on the rectangular coordinate plane, solve systems of them and solve linear programming problems. Each inequality describes the region on one side of a line, and the region where all of them hold is called the feasible region.

Steps for a linear programming problem: define the unknowns; write the constraints as inequalities, including x ≥ 0 and y ≥ 0; draw each boundary line and find the feasible region; solve pairs of boundary equations to find its vertices; then work out the objective function at each vertex, since the greatest and least values occur at vertices. If the answer must be a whole number and a vertex is not, check the nearby whole-number points inside the region.

Example: with x ≥ 0, y ≥ 0, x + y ≤ 6 and x + 2y ≤ 8, find the greatest value of P = 3x + 4y. The vertices of the feasible region are (0, 0), (6, 0), (0, 4) and (4, 2), where x + y = 6 meets x + 2y = 8. P at these points is 0, 18, 16 and 20, so the greatest value is 20, at (4, 2).

Linear programming

Checking habits and how to revise

Most mistakes with inequalities come from careless steps, not hard ideas. Checking each answer in the ways below works better than doing more and more new questions.

Keep a notebook of mistakes: copy out each question where you forgot to flip the sign or mixed up "and" with "or", note why, and redo it later. Parents helping at home need not solve the questions; ask your child to explain each step, especially "did this step multiply or divide by a negative number?". If your child still cannot keep up after class, revise the topic's videos and examples first and then ask a teacher; if you are thinking about tutoring, the tutoring guide explains the options.

Circle the coefficient of x in the last step and check whether it is positive or negative.
Substitute one number inside your answer and one outside into the original inequality: one should work and the other should not.
Check the end points: ≤ and ≥ match solid dots on the number line, < and > match hollow dots.
For compound inequalities, draw both pieces on the number line first, then apply "and" or "or".
For quadratic inequalities, make the x² coefficient positive first, then decide between the roots or outside them.

Frequently asked questions

Why does multiplying by a negative number flip the inequality sign?

Multiplying by a negative number mirrors the whole number line, so a larger number becomes the smaller one: 3 > 2, but −3 < −2. That is why the sign must flip. Adding or subtracting any number, or multiplying or dividing by a positive number, keeps the direction.

How do "and" and "or" differ in compound inequalities?

"And" means both conditions hold at once, so take the overlap on the number line; "or" means either is enough, so take both pieces together. If the two pieces of an "and" inequality do not overlap, there is no solution.

How do I tell whether a quadratic inequality's answer is between the roots or outside them?

First make the x² coefficient positive. The parabola then opens upwards, so "< 0" or "≤ 0" is between the roots and "> 0" or "≥ 0" is outside them. If unsure, test one number from each interval.

Is linear programming tested in the HKDSE?

Linear programming is a non-foundation topic in the senior Compulsory Part. Under the HKEAA assessment framework, Section A of Paper 1 and Paper 2 covers foundation topics only, so linear programming and the algebraic method for quadratic inequalities appear only in Section B. Compound inequalities and the graphical method for quadratic inequalities are foundation topics.

What are the Chinese terms for inequalities?

Inequality is 不等式, a linear inequality in one unknown is 一元一次不等式, a compound inequality is 複合不等式, a quadratic inequality is 一元二次不等式, the feasible region is 可行解區域 and linear programming is 線性規畫. Students who learn maths in both languages should know both sets of terms.

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Secondary 5 maths: inequalities and linear programming