Linear Inequalities

Class 11 · Mathematics

4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.

Inequality Symbols and Intervals

Part of: Inequalities and Solutions

Strict signs exclude endpoints while inclusive signs keep them; open, closed and half-open intervals bracket accordingly; number lines shade with hollow or solid dots.

Why it matters. Interval shorthand compresses solution sets. Dots encode inclusion.

Formulas

The smaller side of an inequality lies left on the number line
[a, b] keeps both ends; (a, b) leaves both ends out

Key words

Inequality: : A statement that two values are not equal, using <, >, <= or >=.
Strict inequality: : A sign pair that excludes its endpoints, like less-than or greater-than.
Closed interval: : An interval including both endpoints, written [a, b].
Open interval: : An interval excluding both endpoints, written (a, b).
Half-open interval: : An interval keeping one endpoint and dropping the other.

Try it in two minutes

Sort strict versus inclusive.
Bracket the interval.
Shade with right dots.

Try at home. Write ages 13 to 19 as intervals.

Solving One-Variable Inequalities

Part of: Inequalities and Solutions

Adding and positive-multiplying preserve order while negative-multiplying flips it; isolate x then intersect with natural, integer or real domains.

Why it matters. Flip rule prevents sign errors. Domains reshape answers.

Formulas

Multiplying or dividing by a negative number flips the inequality sign
Adding the same number to both sides keeps the inequality true

Key words

Linear inequality: : An inequality with constants and x only to the first power.
Solution set: : All values of x that make the inequality true.
Flip rule: : The rule that multiplying by a negative flips the inequality sign.
Replacement set: : The number family, like natural or real, x is chosen from.

Try it in two minutes

Isolate the variable.
Flip on negative multiply.
Intersect with the domain.

Try at home. Solve 3x + 5 < 20 in integers.

Word Problems with Inequalities

Part of: Applications and Systems

Translate at-least and at-most into symbols; consecutive integers offset by fixed steps; mixture and budget constraints bound totals.

Why it matters. Inequalities budget real resources. Translation turns words into maths.

Formulas

at least p means p or more; at most q means q or less
Consecutive integers differ by 1; consecutive evens differ by 2

Key words

At least: : A lower bound p that the value must reach or cross.
At most: : An upper bound q that the value must not cross.
Consecutive integers: : Integers following one another with a fixed gap.
Constraint: : A limiting condition the unknown must satisfy.

Try it in two minutes

Name the unknown.
Translate the bound.
Solve and interpret.

Try at home. Budget weekly snacks under 500.

Systems of Inequalities

Part of: Applications and Systems

Systems need every inequality true at once; solve each apart then overlap intervals on one line; empty overlap means no solution.

Why it matters. Real limits combine. Overlap finds common ground.

Formulas

A system needs every inequality true at the same time
Empty overlap on the number line means no solution

Key words

System of inequalities: : Two or more inequalities solved together for common values.
Common solution: : Values satisfying every inequality of the system at once.
Number-line overlap: : Overlapping the solution lines to read shared values.
No solution: : No value satisfies all inequalities together.

Try it in two minutes

Solve each apart.
Overlap on one line.
Read the common part.

Try at home. Overlap x > 2 with x < 8.

What your child will practise

Inequality Symbols and Intervals: Space Puzzle (Clue Puzzle)
Inequality Symbols and Intervals: Space Quick Fire (Quick Fire)
Solving One-Variable Inequalities: Ocean Puzzle (Clue Puzzle)
Systems of Inequalities: Ocean Quest (Flip and Match)
Word Problems with Inequalities: Candy Quick Fire (Quick Fire)

Quiz: Inequality Symbols and Intervals (8 questions)
Quiz: Solving One-Variable Inequalities (8 questions)
Quiz: Systems of Inequalities (8 questions)
Quiz: Word Problems with Inequalities (8 questions)

Linear Inequalities

Class 11 · Mathematics

4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.

Inequality Symbols and Intervals

Part of: Inequalities and Solutions

Strict signs exclude endpoints while inclusive signs keep them; open, closed and half-open intervals bracket accordingly; number lines shade with hollow or solid dots.

Why it matters. Interval shorthand compresses solution sets. Dots encode inclusion.

Formulas

The smaller side of an inequality lies left on the number line
[a, b] keeps both ends; (a, b) leaves both ends out

Key words

Inequality: : A statement that two values are not equal, using <, >, <= or >=.
Strict inequality: : A sign pair that excludes its endpoints, like less-than or greater-than.
Closed interval: : An interval including both endpoints, written [a, b].
Open interval: : An interval excluding both endpoints, written (a, b).
Half-open interval: : An interval keeping one endpoint and dropping the other.

Try it in two minutes

Sort strict versus inclusive.
Bracket the interval.
Shade with right dots.

Try at home. Write ages 13 to 19 as intervals.

Solving One-Variable Inequalities

Part of: Inequalities and Solutions

Adding and positive-multiplying preserve order while negative-multiplying flips it; isolate x then intersect with natural, integer or real domains.

Why it matters. Flip rule prevents sign errors. Domains reshape answers.

Formulas

Multiplying or dividing by a negative number flips the inequality sign
Adding the same number to both sides keeps the inequality true

Key words

Linear inequality: : An inequality with constants and x only to the first power.
Solution set: : All values of x that make the inequality true.
Flip rule: : The rule that multiplying by a negative flips the inequality sign.
Replacement set: : The number family, like natural or real, x is chosen from.

Try it in two minutes

Isolate the variable.
Flip on negative multiply.
Intersect with the domain.

Try at home. Solve 3x + 5 < 20 in integers.

Word Problems with Inequalities

Part of: Applications and Systems

Translate at-least and at-most into symbols; consecutive integers offset by fixed steps; mixture and budget constraints bound totals.

Why it matters. Inequalities budget real resources. Translation turns words into maths.

Formulas

at least p means p or more; at most q means q or less
Consecutive integers differ by 1; consecutive evens differ by 2

Key words

At least: : A lower bound p that the value must reach or cross.
At most: : An upper bound q that the value must not cross.
Consecutive integers: : Integers following one another with a fixed gap.
Constraint: : A limiting condition the unknown must satisfy.

Try it in two minutes

Name the unknown.
Translate the bound.
Solve and interpret.

Try at home. Budget weekly snacks under 500.

Systems of Inequalities

Part of: Applications and Systems

Systems need every inequality true at once; solve each apart then overlap intervals on one line; empty overlap means no solution.

Why it matters. Real limits combine. Overlap finds common ground.

Formulas

A system needs every inequality true at the same time
Empty overlap on the number line means no solution

Key words

System of inequalities: : Two or more inequalities solved together for common values.
Common solution: : Values satisfying every inequality of the system at once.
Number-line overlap: : Overlapping the solution lines to read shared values.
No solution: : No value satisfies all inequalities together.

Try it in two minutes

Solve each apart.
Overlap on one line.
Read the common part.

Try at home. Overlap x > 2 with x < 8.

What your child will practise

Inequality Symbols and Intervals: Space Puzzle (Clue Puzzle)
Inequality Symbols and Intervals: Space Quick Fire (Quick Fire)
Solving One-Variable Inequalities: Ocean Puzzle (Clue Puzzle)
Systems of Inequalities: Ocean Quest (Flip and Match)
Word Problems with Inequalities: Candy Quick Fire (Quick Fire)

Quiz: Inequality Symbols and Intervals (8 questions)
Quiz: Solving One-Variable Inequalities (8 questions)
Quiz: Systems of Inequalities (8 questions)
Quiz: Word Problems with Inequalities (8 questions)