What is a number line and how is it used in IB MYP Mathematics?

A number line is a visual representation of numbers placed in order on a straight line. In IB MYP Mathematics, it is used to help students understand the concept of order and magnitude of numbers, including integers, fractions, and decimals. It is also a useful tool for visualizing addition, subtraction, and understanding inequalities.

How do you represent inequalities on a number line?

To represent inequalities on a number line, you first identify the critical value(s) from the inequality. Use an open circle to represent a strict inequality (e.g., x > 3 or x < 3) and a closed circle for inclusive inequalities (e.g., x ≥ 3 or x ≤ 3). Then, shade the portion of the line that represents all possible solutions to the inequality.

What are the common types of inequalities studied in the IB MYP curriculum?

In the IB MYP curriculum, students commonly study linear inequalities, which involve expressions like ax + b > c. They also explore compound inequalities, which combine two inequalities, and absolute value inequalities, which involve expressions like |x| < a. These help students understand how to solve and graph inequalities on a number line.

How can solving inequalities be different from solving equations?

Solving inequalities is similar to solving equations, but there are key differences. When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed. This is not the case with equations. Additionally, inequalities often have a range of solutions, which can be represented on a number line, unlike equations that typically have a single solution.

Why is understanding inequalities important in real-world applications?

Understanding inequalities is crucial for solving real-world problems where conditions are not exact but fall within a range. For instance, inequalities are used in budgeting, where expenses must not exceed income, or in engineering, where materials must withstand certain stress levels. In the IB MYP Mathematics curriculum, students learn to apply these concepts to practical scenarios, enhancing their problem-solving skills.

Why is "Forgetting to flip the sign when dividing by a negative." a common mistake in Number Line and Inequality?

Why it happens: Treating it like an equation. How to avoid it: Mantra: 'Times or divide by NEGATIVE? FLIP THE SIGN.' Always check by substituting a value.

Why is "Using closed circle for $ $." a common mistake in Number Line and Inequality?

Why it happens: Habit. How to avoid it: are STRICT — open (hollow) circle. Only and get the closed (filled) circle.

Why is "Flipping the sign when adding a negative." a common mistake in Number Line and Inequality?

Why it happens: Confusion with the multiply rule. How to avoid it: Adding or subtracting NEVER flips. Only × or ÷ by a negative flips.

NumberIB MYP Maths Extended Level

Number Line and Inequality

Q: What are the common types of inequalities studied in the IB MYP curriculum?

In the IB MYP curriculum, students commonly study linear inequalities, which involve expressions like ax + b > c. They also explore compound inequalities, which combine two inequalities, and absolute value inequalities, which involve expressions like |x| < a. These help students understand how to solve and graph inequalities on a number line.

Q: How can solving inequalities be different from solving equations?

Solving inequalities is similar to solving equations, but there are key differences. When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed. This is not the case with equations. Additionally, inequalities often have a range of solutions, which can be represented on a number line, unlike equations that typically have a single solution.

Q: Why is understanding inequalities important in real-world applications?

Understanding inequalities is crucial for solving real-world problems where conditions are not exact but fall within a range. For instance, inequalities are used in budgeting, where expenses must not exceed income, or in engineering, where materials must withstand certain stress levels. In the IB MYP Mathematics curriculum, students learn to apply these concepts to practical scenarios, enhancing their problem-solving skills.

Q: Why is "Forgetting to flip the sign when dividing by a negative." a common mistake in Number Line and Inequality?

Why it happens: Treating it like an equation. How to avoid it: Mantra: 'Times or divide by NEGATIVE? FLIP THE SIGN.' Always check by substituting a value.

Q: Why is "Using closed circle for $ $." a common mistake in Number Line and Inequality?

Why it happens: Habit. How to avoid it: are STRICT — open (hollow) circle. Only and get the closed (filled) circle.

Q: Why is "Flipping the sign when adding a negative." a common mistake in Number Line and Inequality?

Why it happens: Confusion with the multiply rule. How to avoid it: Adding or subtracting NEVER flips. Only × or ÷ by a negative flips.

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Number Line and Inequality — IB MYP Mathematics (Extended): solving and graphing inequalities

Inequalities describe ranges of values, not single points. This MYP Mathematics Extended note covers solving inequalities, graphing on a number line, and the crucial sign-flip rule.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Solve linear inequalities.
MYP Mathematics A — Represent solutions on a number line.
MYP Mathematics C — Use correct inequality notation including open / closed circles.
MYP Mathematics D — Apply inequalities to real-life constraints.

Inequality symbols and number line

Four symbols, two types of endpoint.

Symbol	Meaning	Endpoint on number line
$<$	less than	OPEN circle (hollow)
$>$	greater than	OPEN circle (hollow)
$\le$	less than or equal to	CLOSED circle (filled)
$\ge$	greater than or equal to	CLOSED circle (filled)

Use:

OPEN $\circ$ when the endpoint is NOT included.
CLOSED $\bullet$ when the endpoint IS included.

The inequality $-1 < x \le 3$. Open circle at $-1$ (strictly greater); closed at $3$ (less than or equal). The shaded line in between is the solution.

Compound inequalities combine two conditions:

$-1 < x \le 3$ : $x$ is strictly greater than $-1$ AND less than or equal to $3$ .

Interval notation is an alternative:

$(a, b)$ : open interval, neither endpoint included.
$[a, b]$ : closed interval, both endpoints included.
$[a, b)$ : closed on left, open on right.
$(-\infty, 3]$ : all numbers $\le 3$ . Infinity is always open (you can't 'reach' it).

$<, >$ : open circle.
$\le, \ge$ : closed circle.
Shaded line shows the solution range.
Interval notation: $[\,]$ closed, $(\,)$ open.

Solving inequalities

Like solving equations — but watch the sign-flip rule.

Most steps in solving an inequality are IDENTICAL to solving an equation:

Add or subtract the same value from both sides.
Multiply or divide both sides by the same POSITIVE value.

One vital exception: when you MULTIPLY or DIVIDE both sides by a NEGATIVE number, you MUST FLIP the inequality sign.

Worked example. Solve $2x - 5 < 7$ .

Add 5 to both sides: $2x < 12$ .
Divide by 2 (positive — no flip): $x < 6$ .

Worked example. Solve $-3x + 4 \ge 10$ .

Subtract 4: $-3x \ge 6$ .
Divide by $-3$ . NEGATIVE — FLIP: $x \le -2$ .

Worked example (compound). Solve $-7 \le 3x + 2 < 8$ .

Treat as a single inequality with two operations applied on ALL THREE parts:

Subtract 2: $-9 \le 3x < 6$ .
Divide by 3 (positive): $-3 \le x < 2$ .

On a number line: closed circle at $-3$ , open circle at $2$ , shaded between.

Why does the sign flip? Multiplying by a negative reflects the number line through 0. The order of numbers reverses: $2 > 1$ becomes $-2 < -1$ . So an inequality must also flip.

$+, -$ : NEVER flip.
$\times, \div$ by positive: NEVER flip.
$\times, \div$ by NEGATIVE: ALWAYS flip.
Compound: apply to all three parts.

Real-life applications

Inequalities model constraints.

Many real-life situations are best modelled as inequalities, not equations:

Speed limits. $\text{speed} \le 80\,\text{km/h}$ .
Age requirements. $\text{age} \ge 16$ to drive a moped (in some countries).
Capacity. A lift with 'max 8 people' means $n \le 8$ .
Budget. 'I can spend at most £50' means $\text{total} \le 50$ .

Worked example. A taxi charges £3.50 plus £1.20 per km. You have £20. What is the maximum distance you can travel?

Total cost $= 3.50 + 1.20d$ , where $d$ is distance in km.
$3.50 + 1.20d \le 20$ .
$1.20d \le 16.5$ .
$d \le 13.75\,\text{km}$ .

So you can travel up to $13.75\,\text{km}$ . In a practical context, you'd round DOWN to $13.75$ (or $13\,\text{km}$ if the taxi rounds whole km).

For criterion D, always:

Define your variable clearly.
Set up the inequality.
Solve.
Interpret in context (units, sensible rounding).

Inequality models a constraint.
Define the variable in WORDS.
Solve algebraically.
Interpret in context: units, sensible rounding.

How it’s examined

Criterion A: solve linear and compound inequalities. Criterion C: clear number-line representations. Criterion D: model a constraint as an inequality.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — Number Line and Inequality

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

1Solve a linear inequality
Getting started• linear
Question
Solve $3x - 7 < 11$ and represent the solution on a number line.
Step-by-step solution
1. Step 1
  Add 7 to both sides: $3x < 18$ .
2. Step 2
  Divide by 3 (positive, no flip): $x < 6$ .
3. Step 3
  Number line: open circle at 6, shaded line extending to the LEFT (towards $-\infty$ ).
Answer
$x < 6$
2Negative coefficient — flip the sign
Building confidence• sign flip
Question
Solve $-2x + 5 \le 1$ .
Step-by-step solution
1. Step 1
  Subtract 5: $-2x \le -4$ .
2. Step 2
  Divide by $-2$ . NEGATIVE — FLIP the sign: $x \ge 2$ .
3. Step 3
  Check: $x = 2 \Rightarrow -2(2) + 5 = 1 \le 1$ . ✓
Answer
$x \ge 2$
3Compound inequality
Building confidence• compound
Question
Solve $-5 \le 2x + 3 < 9$ .
Step-by-step solution
1. Step 1
  Subtract 3 from ALL three parts: $-8 \le 2x < 6$ .
2. Step 2
  Divide all three by 2 (positive, no flip): $-4 \le x < 3$ .
3. Step 3
  Number line: closed at $-4$ , open at $3$ , shaded between.
Answer
$-4 \le x < 3$
4Maximum number of items
Stretch• application
Question
A taxi charges $4 plus $1.50 per km. You have $25. What is the maximum number of WHOLE km you can travel?
Step-by-step solution
1. Step 1
  Let $d$ = distance in km. Total cost = $4 + 1.5d$ .
2. Step 2
  Inequality: $4 + 1.5d \le 25$ .
3. Step 3
  Subtract 4: $1.5d \le 21$ .
4. Step 4
  Divide by 1.5: $d \le 14$ .
5. Step 5
  In context: whole km, so maximum 14 km.
Answer
$d \le 14$ km — max 14 whole km.

Key Definitions and Keywords — Number Line and Inequality

Definitions to memorise and the exact keywords mark schemes credit for number line and inequality answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Inequality
Examiner keyword
A statement that one quantity is less than (or greater than) another, possibly with equality.
Strict inequality
$<$ or $>$ — does not include equality.
Non-strict (weak) inequality
$\le$ or $\ge$ — includes equality.
Compound inequality
Examiner keyword
Two inequalities combined: e.g. $-3 \le x < 5$ .

Common Mistakes and Misconceptions — Number Line and Inequality

The traps other students keep falling into on number line and inequality questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Forgetting to flip the sign when dividing by a negative.
Why it happens
Treating it like an equation.
How to avoid it
Mantra: 'Times or divide by NEGATIVE? FLIP THE SIGN.' Always check by substituting a value.
✕Using closed circle for $<$ or $>$ .
Why it happens
Habit.
How to avoid it
$<$ and $>$ are STRICT — open (hollow) circle. Only $\le$ and $\ge$ get the closed (filled) circle.
✕Flipping the sign when adding a negative.
Why it happens
Confusion with the multiply rule.
How to avoid it
Adding or subtracting NEVER flips. Only $\times$ or $\div$ by a negative flips.

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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Number Line and Inequality — frequently asked questions

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Number Line and Inequality

Short Study Notes in the form of Slides

Short Study Notes — Number Line and Inequality

Detailed Notes

What you’ll learn

How it’s examined

1Solve a linear inequality

2Negative coefficient — flip the sign

3Compound inequality

4Maximum number of items

Inequality

Strict inequality

Non-strict (weak) inequality

Compound inequality

✕Forgetting to flip the sign when dividing by a negative.

✕Using closed circle for <<< or >>>.

✕Flipping the sign when adding a negative.

Practice questions

Past paper style quiz

Assess your understanding

Number Line and Inequality — frequently asked questions

✕Using closed circle for $<$ or $>$ .