What is an inequality in Algebra and how is it different from an equation?

In Algebra, an inequality is a mathematical statement that compares two expressions using inequality symbols such as , ≤, or ≥. Unlike equations, which show that two expressions are equal, inequalities show that one expression is greater or less than the other. For example, x > 3 means that x can be any number greater than 3.

How do you solve a linear inequality in one variable?

To solve a linear inequality in one variable, follow similar steps as solving a linear equation: simplify both sides, isolate the variable on one side, and solve for the variable. Remember to reverse the inequality sign if you multiply or divide by a negative number. For example, to solve 2x - 3 > 5, add 3 to both sides to get 2x > 8, then divide by 2 to find x > 4.

What are the common mistakes to avoid when solving inequalities?

Common mistakes include forgetting to reverse the inequality sign when multiplying or dividing by a negative number, and incorrectly combining terms. It's also important to check your solution by substituting values back into the original inequality to ensure they satisfy the condition.

How can you graph the solution of an inequality on a number line?

To graph the solution of an inequality on a number line, first solve the inequality for the variable. Then, draw a number line and use an open circle to represent a value that is not included in the solution (for ) or a closed circle for values that are included (for ≤ or ≥). Shade the region of the number line that represents all possible solutions. For example, for x > 2, place an open circle at 2 and shade to the right.

What is a compound inequality and how do you solve it?

A compound inequality consists of two or more inequalities joined by 'and' or 'or'. To solve a compound inequality, solve each inequality separately and then find the intersection (for 'and') or union (for 'or') of the solutions. For example, for 1 1 gives x > -1, and x + 2 ≤ 5 gives x ≤ 3. The solution is -1 < x ≤ 3.

Why is "Solving the quadratic as if it were an equation." a common mistake in Inequality?

Why it happens: Forgetting to set up the inequality. How to avoid it: First move EVERYTHING to one side (with > 0, < 0, 0, or 0). Then factorise. Then choose regions.

Why is "Choosing the wrong region of the parabola." a common mistake in Inequality?

Why it happens: Confusion about 'inside' vs 'outside'. How to avoid it: For an UPWARD parabola: > 0 outside the roots, < 0 between. For DOWNWARD: reverse. ALWAYS TEST a value in each region to be sure.

Why is "Forgetting to flip the sign when multiplying inequality by a negative." a common mistake in Inequality?

Why it happens: Same as for linear inequalities. How to avoid it: Same rule: × or ÷ by negative → FLIP.

AlgebraIB MYP Maths Extended Level

Inequality

Q: Why is "Solving the quadratic as if it were an equation." a common mistake in Inequality?

Why it happens: Forgetting to set up the inequality. How to avoid it: First move EVERYTHING to one side (with > 0, < 0, 0, or 0). Then factorise. Then choose regions.

Q: Why is "Choosing the wrong region of the parabola." a common mistake in Inequality?

Why it happens: Confusion about 'inside' vs 'outside'. How to avoid it: For an UPWARD parabola: > 0 outside the roots, < 0 between. For DOWNWARD: reverse. ALWAYS TEST a value in each region to be sure.

Q: Why is "Forgetting to flip the sign when multiplying inequality by a negative." a common mistake in Inequality?

Why it happens: Same as for linear inequalities. How to avoid it: Same rule: × or ÷ by negative → FLIP.

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Inequality (Algebra) — IB MYP Mathematics (Extended): linear and quadratic inequalities

We extend the inequality rules from numbers to algebraic expressions. This MYP Mathematics Extended note covers linear inequalities (recap with the sign-flip rule) and quadratic inequalities (parabola sign analysis).

What you’ll learn

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

MYP Mathematics A — Solve linear inequalities.
MYP Mathematics A — Solve quadratic inequalities using sign analysis.
MYP Mathematics C — Express solutions as inequalities or intervals.
MYP Mathematics D — Apply inequalities to optimisation contexts.

Linear inequalities (recap)

Solve like equations; FLIP when multiplying/dividing by negative.

Linear inequalities involve $ax + b < c$ (or $\le, >, \ge$ ). Most rules match those for equations:

Add or subtract any value from both sides — no flip.
Multiply or divide by a POSITIVE — no flip.
Multiply or divide by a NEGATIVE — FLIP the inequality.

Worked example. Solve $4 - 3x \ge 13$ .

Subtract 4: $-3x \ge 9$ .
Divide by $-3$ , FLIP: $x \le -3$ .

Worked example (compound). Solve $-5 < 2x - 1 \le 7$ .

Add 1 to all three parts: $-4 < 2x \le 8$ .
Divide by 2: $-2 < x \le 4$ .

Interval notation: $(-2, 4]$ .

Add/subtract: no flip.
Multiply/divide by NEGATIVE: FLIP.
Compound: apply operations to ALL three parts.

Quadratic inequalities

Factorise, find roots, decide sign in each region.

A quadratic inequality has the form $ax^2 + bx + c > 0$ (or $<, \le, \ge$ ).

Strategy:

Move everything to one side, with $>$ or $<$ 0.
Factorise.
Find the critical values (where the quadratic = 0).
Test a value in each region; record the sign.
Read off the solution from the regions matching the inequality.

Worked example. Solve $x^2 - 5x + 6 > 0$ .

Factorise: $(x - 2)(x - 3) > 0$ .
Critical values: $x = 2$ and $x = 3$ .
Three regions: $x < 2$ , $2 < x < 3$ , $x > 3$ .
Test each:
- $x = 0$ (in region 1): $(0-2)(0-3) = (-)(-) = (+)$ . ✓
- $x = 2.5$ (in region 2): $(0.5)(-0.5) = (-)$ . ✗
- $x = 4$ (in region 3): $(2)(1) = (+)$ . ✓
Solution: $x < 2$ or $x > 3$ .

The parabola sits ABOVE the x-axis in the green regions (positive). Solution to $(x-2)(x-3) > 0$ is $x < 2$ or $x > 3$.

Worked example. Solve $x^2 - 5x + 6 \le 0$ (the opposite inequality).

Same critical values, but the parabola is $\le 0$ BETWEEN the roots.
Solution: $2 \le x \le 3$ .

Pattern:

For an UPWARD parabola: $> 0$ → outside the roots; $< 0$ → between the roots.
For a DOWNWARD parabola (negative $a$ ): reverse.

Factorise; find roots; test each region.
Upward parabola, $> 0$ : outside roots.
Upward parabola, $< 0$ : between roots.
Use $\le$ or $\ge$ to include the endpoints.

Combined and applied inequalities

Real-world constraints often combine linear + quadratic.

Worked example (with absolute value, Extended). Solve $|x - 4| < 3$ .

Translate: $-3 < x - 4 < 3$ .
Add 4: $1 < x < 7$ .

Worked example (real context). A car park charges $5 plus $2 per hour. You can spend at most $20. For how many hours can you park?

$5 + 2h \le 20$ .
$h \le 7.5$ hours.

Worked example (quadratic context). The height of a ball $h$ metres after $t$ seconds is $h = 20t - 5t^2$ . For what time interval is the ball ABOVE 15 m?

$20t - 5t^2 > 15$ .
Rearrange: $5t^2 - 20t + 15 < 0 \Rightarrow t^2 - 4t + 3 < 0$ .
Factorise: $(t - 1)(t - 3) < 0$ .
Upward parabola, $< 0$ between roots: $1 < t < 3$ .
So the ball is above 15 m between 1 and 3 seconds.

Real contexts (criterion D) often need you to interpret the solution: 'so the ball is above 15 m for 2 seconds, between $t = 1$ and $t = 3$ '.

Real-context inequalities: define variable, set up inequality, solve, interpret.
Absolute value: $|x| < a \iff -a < x < a$ .
Quadratic in real contexts: read solution as a time / range / domain.

How it’s examined

Criterion A: solve linear and quadratic inequalities. Criterion C: state solutions clearly with intervals. Criterion D: model and solve real-world inequalities.

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

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Step-by-step worked examples — Inequality

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

1Solve a linear inequality
Getting started• linear
Question
Solve $5 - 2x > 13$ .
Step-by-step solution
1. Step 1
  Subtract 5: $-2x > 8$ .
2. Step 2
  Divide by $-2$ , FLIP: $x < -4$ .
Answer
$x < -4$
2Quadratic inequality (greater than)
Building confidence• quadratic
Question
Solve $x^2 - 4x - 5 > 0$ .
Step-by-step solution
1. Step 1
  Factorise: $(x - 5)(x + 1) > 0$ .
2. Step 2
  Critical values: $x = 5$ and $x = -1$ .
3. Step 3
  Upward parabola, $> 0$ : OUTSIDE the roots.
Answer
$x < -1$ or $x > 5$ .
3Quadratic inequality (less than)
Building confidence• quadratic
Question
Solve $x^2 + 2x - 8 \le 0$ .
Step-by-step solution
1. Step 1
  Factorise: $(x + 4)(x - 2) \le 0$ .
2. Step 2
  Critical values: $x = -4$ and $x = 2$ .
3. Step 3
  Upward parabola, $\le 0$ : BETWEEN the roots, including endpoints.
Answer
$-4 \le x \le 2$ .
4Real context (Extended)
Stretch• application
Question
A rectangular garden has length $(x + 5)$ m and width $(x - 2)$ m. For what values of $x$ is the area more than 30 m²?
Step-by-step solution
1. Step 1
  Need $x - 2 > 0$ (positive width), so $x > 2$ .
2. Step 2
  Set up inequality: $(x+5)(x-2) > 30$ .
3. Step 3
  Expand: $x^2 + 3x - 10 > 30 \Rightarrow x^2 + 3x - 40 > 0$ .
4. Step 4
  Factorise: $(x + 8)(x - 5) > 0$ . Critical values $x = -8, x = 5$ .
5. Step 5
  Upward parabola, $> 0$ : outside the roots. So $x < -8$ or $x > 5$ . Combined with $x > 2$ : $x > 5$ .
Answer
$x > 5$ (the length must exceed 10 m, width 3 m).

Key Definitions and Keywords — Inequality

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

Quadratic inequality
Examiner keyword
An inequality of the form $ax^2 + bx + c > 0$ (or $<, \le, \ge$ ).
Critical values
Examiner keyword
The values where the expression equals zero — the roots of the quadratic.
Sign analysis
Determining the sign of an expression in each region between critical values.

Common Mistakes and Misconceptions — Inequality

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

✕Solving the quadratic as if it were an equation.
Why it happens
Forgetting to set up the inequality.
How to avoid it
First move EVERYTHING to one side (with $> 0$ , $< 0$ , $\ge 0$ , or $\le 0$ ). Then factorise. Then choose regions.
✕Choosing the wrong region of the parabola.
Why it happens
Confusion about 'inside' vs 'outside'.
How to avoid it
For an UPWARD parabola: $> 0$ outside the roots, $< 0$ between. For DOWNWARD: reverse. ALWAYS TEST a value in each region to be sure.
✕Forgetting to flip the sign when multiplying inequality by a negative.
Why it happens
Same as for linear inequalities.
How to avoid it
Same rule: $\times$ or $\div$ by negative → FLIP.

Practice questions

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Past paper style quiz

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Inequality — frequently asked questions

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Inequality

Short Study Notes in the form of Slides

Short Study Notes — Inequality

Detailed Notes

What you’ll learn

How it’s examined

1Solve a linear inequality

2Quadratic inequality (greater than)

3Quadratic inequality (less than)

4Real context (Extended)

Quadratic inequality

Critical values

Sign analysis

✕Solving the quadratic as if it were an equation.

✕Choosing the wrong region of the parabola.

✕Forgetting to flip the sign when multiplying inequality by a negative.

Practice questions

Past paper style quiz

Assess your understanding

Inequality — frequently asked questions