🔮 Algebra Realm · Equations

Inequalities

Inequalities focuses on how to compare ranges of possible values using inequality symbols instead of a single equals sign. In this lesson, focus on substitution replaces a letter with a known value.

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Understand Inequalities

Inequalities focuses on how to compare ranges of possible values using inequality symbols instead of a single equals sign. In this lesson, focus on substitution replaces a letter with a known value.

Substitution replaces a letter with a known value. Brackets protect negative or fractional substitutions, and evaluating both sides independently checks whether a proposed solution is valid. For inequalities, the final written answer should make that exact relationship visible rather than hiding it inside an unexplained result.

Start here

Inequalities: Substitution replaces a letter with a known value. Repeat the evaluation with explicit brackets and inspect each intermediate value. Keep the inequalities representation visible until the final line.

Picture the idea

Drop a value into each variable slot and watch a calculation tree evaluate one operation at a time. Use the model to explain one change you notice while working on inequalities.

Check as you go

Repeat the evaluation with explicit brackets and inspect each intermediate value. Write that check beside the final inequalities answer.

Key vocabulary

Rules and key facts

x < 5 allows 4 but not 5.

Write the expression with the replacement in brackets.
Follow the operation order carefully.
Evaluate each side independently if checking an equation.
Compare the results and state the conclusion. Record the check explicitly for inequalities.

Step-by-step method

Write the expression with the replacement in brackets.
Follow the operation order carefully.
Evaluate each side independently if checking an equation.
Compare the results and state the conclusion. Record the check explicitly for inequalities.

What you need first

Recognise the vocabulary: substitute, evaluate, expression.
Be able to explain the purpose of inequalities before calculating.
Keep the relevant values, units and representation visible while you work.

Real-world use

Formula use
Checking models

Visual / interactive

See the idea, then move it around

Skip to Practice

Drop a value into each variable slot and watch a calculation tree evaluate one operation at a time. Use the model to explain one change you notice while working on inequalities.

Interactive maths model Connected to this topic; move controls, check outputs, then earn XP only from verified actions.

Responsive · validated · topic linked

Worked examples

Examples, methods and exam thinking

Level 1 · Foundation

Understand the idea with small numbers, one representation and one clear step.

Level 2 · Secure

Use the standard Year 8 method with mixed examples and normal wording.

Level 3 · Challenge

Handle multi-step or less familiar questions and explain choices.

Level 4 · Exam-style

Solve a worded question, show reasoning, check accuracy and write a final sentence.

Foundation example

Build confidence

Given information: Inequalities — Find 3x + 2 when x = 2. Method choice: use the inequalities method and show each step with the stated values. Calculation or reasoning: Substitute x = 2: 3 × 2 + 2 = 8. Final answer: 8. Check: substitute or compare with the original information to confirm the result fits the question.

Write the expression with the replacement in brackets.
Follow the operation order carefully.
Evaluate each side independently if checking an equation.

Secure example

Use the normal method

Given information: Inequalities — Find 3x + 14 when x = 10. Method choice: use the inequalities method and show each step with the stated values. Calculation or reasoning: Substitute x = 10: 3 × 10 + 14 = 44. Final answer: 44. Check: substitute or compare with the original information to confirm the result fits the question.

Check: Draw an open or filled circle and shade the correct direction.

Challenge example

Stretch the idea

Given information: Inequalities — Find 3x + 13 when x = 7. Method choice: use the inequalities method and show each step with the stated values. Calculation or reasoning: Substitute x = 7: 3 × 7 + 13 = 34. Final answer: 34. Check: substitute or compare with the original information to confirm the result fits the question.

Try explaining why each step works before checking the answer.

Exam-style example

Show your reasoning

Given information: Inequalities — Find 3x + 12 when x = 4. Method choice: use the inequalities method and show each step with the stated values. Calculation or reasoning: Substitute x = 4: 3 × 4 + 12 = 24. Final answer: 24. Check: substitute or compare with the original information to confirm the result fits the question.

Common mistakes

Dropping brackets around negative values. This is a key trap when answering inequalities questions.
Assuming a proposed answer is correct before evaluating both sides.

How to check your answer

Repeat the evaluation with explicit brackets and inspect each intermediate value. Write that check beside the final inequalities answer.

Extension challenge

Create a inequalities problem with a tempting incorrect answer. Solve it, apply the check, and explain exactly where the incorrect method breaks down.

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Inequalities challenge

Use substitution lock controls to solve three checked inequalities rounds. Solve at least two of three marked rounds and use feedback to correct any error.

Press Start Game to enter a topic-specific maths arena.

Boss challenge

Challenge Inequalities Arcane Boss

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Inequalities Arcane Boss

Timed mixed-difficulty battle. Practice first if you want, or jump in and learn from feedback.

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Study cards and flashcards · always open

Study the essentials quickly

Study cards

Core idea

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Key vocabulary

substitute · evaluate · expression · solution · check · inequalities

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Rules

Write the expression with the replacement in brackets. Follow the operation order carefully. Evaluate each side independently if checking an equation. Compare the results and state the conclusion. Record the check explicitly for inequalities.

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Formula / fact

x < 5 allows 4 but not 5.

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Foundation example

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Secure example

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Challenge example

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Exam-style example

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Common mistake

Dropping brackets around negative values. This is a key trap when answering inequalities questions.

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Exam tip

For inequalities, show the key representation before the final calculation. Use this final check: Repeat the evaluation with explicit brackets and inspect each intermediate value.

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Real-world use

Formula use, Checking models

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Checklist

I can explain inequalities, use the method, check for mistakes, and answer an exam-style question.

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Flashcards

I’m Stuck

Help for Inequalities

Use this whenever a question feels confusing. Nothing here is locked.

Simple explanation

Think of inequalities as a careful model: make the important values visible, change one thing at a time, and use the check to prove the answer fits.

Step-by-step breakdown

Write the expression with the replacement in brackets.
Follow the operation order carefully.
Evaluate each side independently if checking an equation.
Compare the results and state the conclusion. Record the check explicitly for inequalities.

Hint 1

Start by naming the given information and the exact result required for inequalities.

Hint 2

Write the expression with the replacement in brackets.

Full worked solution

Method: Write the expression with the replacement in brackets. → Follow the operation order carefully. → Evaluate each side independently if checking an equation. → Compare the results and state the conclusion. Record the check explicitly for inequalities.

Common mistake warning

Dropping brackets around negative values. This is a key trap when answering inequalities questions.

Choose a support button above when you need a nudge.

Mastery milestones

Badges reward learning, not locked clicking

I can explain inequalities in my own words.
I can use these words accurately: substitute, evaluate, expression.
I can follow the 4-step method without guessing.
I can avoid this mistake: Dropping brackets around negative values.
I can apply this check: Repeat the evaluation with explicit brackets and inspect each intermediate value.

🥉 Bronze Badge Foundation level completed

🥈 Silver Badge Apprentice level completed

🥇 Gold Badge Skilled level completed

💎 Platinum Badge Mastery test passed

🐉 Legendary Badge Boss battle defeated