Algebra

Linear Equations & Inequalities

Solve Linear Equations

Circle the correct solution.

2x + 7 = 15

x = 4

x = 11

x = 7

3y − 5 = 13

y = 6

y = 8

y = 3

x/3 + 4 = 7

x = 9

x = 11

x = 3

5(n − 2) = 20

n = 6

n = 4

n = 2

2x + 3 = x + 7

x = 4

x = 10

x = 5

Match Equation to Solution

Draw a line to match each equation to its solution.

4x − 3 = 17

3(x + 2) = 18

x/2 − 1 = 5

2x + 5 = x + 9

x = 12

x = 4

x = 5

Inequality Symbols

Match each word description to the correct inequality symbol.

x is at most 5

x is greater than 3

x is less than 10

x is at least 7

< (less than)

≤ (at most)

> (greater than)

≥ (at least)

Solve Inequalities

Circle the correct solution.

x + 4 > 9

x > 5

x > 13

x < 5

2x ≤ 14

x ≤ 7

x ≤ 12

x ≥ 7

3x − 6 < 12

x < 6

x < 2

x > 6

x/4 ≥ 3

x ≥ 12

x ≥ 7

x ≤ 12

Equations with Fractions

Circle the correct solution.

x/2 = 7

x = 14

x = 3.5

x = 9

3x/4 = 9

x = 12

x = 6.75

x = 36

2(x + 3)/5 = 4

x = 7

x = 2

x = 10

Match Situation to Inequality

Draw a line to match each situation to the correct inequality.

You need more than 60 points to pass

The bag can carry at most 15 kg

You must be at least 18 years old

Temperature must be below 5°C for frost

T < 5

w ≤ 15

a ≥ 18

p > 60

Word Problems

Write an equation or inequality, then solve.

Three times a number, minus 8, equals 22. Find the number.

You have $50 and spend $x on a book. You want more than $15 left. Write and solve an inequality.

Check whether x = −2 is a solution to 3x + 11 > 4. Show your substitution.

Equations with Decimals

Solve each equation. Circle the correct answer.

0.5x + 3 = 5.5

x = 5

x = 8.5

x = 2.5

2.4x − 1.2 = 6

x = 3

x = 2

x = 4.5

0.1x = 4.5

x = 45

x = 0.45

x = 450

Simultaneous Equations — Guess and Check

Two equations must both be true at the same time. Try the given values.

x + y = 10 and x − y = 2. Try x = 6, y = 4:

Both true ✓

Only first true

Neither true

2x + y = 11 and x + y = 7. Try x = 4, y = 3:

Both true ✓

Only first true

Neither true

3x − y = 5 and x + y = 7. Try x = 3, y = 4:

Both true ✓

Only second true

Neither true

Inequality Direction — Flip or Not?

When solving inequalities, the sign flips ONLY when multiplying or dividing by a negative number. Sort each step.

Add 5 to both sides

Multiply both sides by −2

Divide both sides by 3

Divide both sides by −4

Subtract 7 from both sides

Multiply both sides by −1

Sign stays the same

Sign flips

Inequalities in Context

Write the inequality, solve it and describe what the answer means.

A lift can carry at most 500 kg. Each person weighs an average of 75 kg. Write and solve an inequality to find the maximum number of people. Explain your answer.

Solve −3x + 7 ≥ 1. Show all steps and explain why the inequality sign changes (or stays the same) at each step.

Equation or Inequality?

Draw a line to match each situation to the correct mathematical statement.

A number is exactly double another

Height must exceed 120 cm to ride

Total cost equals $45

Speed must not exceed 60 km/h

s ≤ 60

3x = 45

h > 120

y = 2x

Equations and Inequalities at Home

Find equations and inequalities in real life.

1Look at a recipe for 4 people. Write an equation to find how much of one ingredient you need for 10 people. Solve it.
2Check a speed limit sign or a weight limit on a bridge. Write this as an inequality using appropriate symbols.
3Think of a budget: you have $80 for a week of lunches. Write an inequality for the maximum cost per day if you buy lunch 5 days. Solve and explain.

One-Step Equations

Solve each equation in one step.

x + 8 = 15

x = 7

x = 23

x = 8

y − 4 = 9

y = 13

y = 5

y = 36

3n = 24

n = 8

n = 27

n = 21

m/5 = 7

m = 35

m = 12

m = 2

Two-Step Equations

Solve each equation in two steps.

2x + 3 = 11

x = 4

x = 7

x = 5

3y − 7 = 14

y = 7

y = 2⅓

y = 5

n/4 + 2 = 6

n = 16

n = 2

n = 32

5m − 3 = 17

m = 4

m = 2.8

m = 20

Equations with Fractions

Solve each equation. Watch for fraction arithmetic.

x/3 = 5

x = 15

x = 8/3

x = 2

2x/5 = 4

x = 10

x = 8/5

x = 20

x/2 + x/3 = 5

x = 6

x = 10

x = 15

(x + 1)/4 = 3

x = 11

x = 3

x = 7

Equation from Word Problems

Write an equation and solve it. Show full working.

I think of a number. I multiply it by 4 and add 7. The result is 31. Find the number.

The perimeter of a rectangle is 54 cm. The length is 3 cm more than twice the width. Find the dimensions.

Tickets to a show cost $12 for adults and $7 for children. A family spends $73 on tickets. If there are 2 adults, how many children attended?

Solving Inequalities — Sign Change

In which step does the inequality sign need to flip?

Dividing both sides of −2x > 8 by −2:

Sign flips: x < −4

Sign stays: x > −4

Sign flips: x > 4

Multiplying both sides of x < 3 by −1:

Sign flips: −x > −3

Sign stays: −x < −3

Sign flips: x > −3

Adding 5 to both sides of x − 5 > 2:

Sign stays: x > 7

Sign flips: x < 7

Undefined

Solving Inequalities with Checking

Solve each inequality and check your answer with a test value.

Solve: 2x − 5 > 7. Test x = 7 in the original inequality to confirm.

Solve: −4x + 3 ≥ −9. Show all steps, naming when and why the sign changes.

Solve: 3(x − 2) ≤ 2(x + 1). Expand first, then solve.

Forming and Solving Inequalities

Write an inequality, solve it, and interpret the solution.

You have $120 in savings. Each week you earn $45. You want at least $300 total. Write an inequality and find how many weeks you need to save.

A rectangular garden must have a perimeter of no more than 60 m. The width is fixed at 12 m. Write an inequality for the maximum length and solve it.

Equation, Inequality, or Identity?

Sort each statement.

2x + 3 = 11

3y − 1 > 5

x + x = 2x

5m ≤ 30

a(b + c) = ab + ac

n/2 = 4

Equation (one specific solution)

Inequality (range of solutions)

Identity (true for all values)

Multi-Step Equation and Inequality Challenge

Solve each problem, showing every algebraic step.

Solve: 3(2x − 1) − 2(x + 4) = 5(x − 2). Show every step.

Solve: 2(3x + 1)/5 ≤ (x + 4)/2. (Multiply both sides by the LCM of the denominators first.)

TipThese are more challenging than typical Year 8 problems — they are suitable for a student who wants to be prepared for Year 9 algebra.

Checking Solutions to Inequalities

Substitute the given value and check whether it satisfies the inequality.

Does x = 4 satisfy 3x − 2 > 10?

Yes: 10 > 10 (No, it's not strictly greater)

No: 12 − 2 = 10, which is not > 10

Yes: 12 > 10 ✓

Does x = −3 satisfy 2x + 5 ≤ 0?

Yes: −6 + 5 = −1 ≤ 0 ✓

No: −6 + 5 = 1 > 0

Cannot tell

Does x = 2 satisfy 4 < 3x − 1 < 10?

No: 3(2)−1 = 5, but 4 < 5 < 10 ✓ so YES

No: 5 is not between 4 and 10

Cannot tell

Investigation: When Does the Inequality Sign Flip?

Investigate the rule about flipping the inequality sign.

Start with 2 < 6. Add 3 to both sides. Subtract 5. Multiply by 2. Does the inequality hold? Multiply by −1. What happens?

Explain in your own words: when does the inequality sign flip, and why does this make sense geometrically (think about a number line)?

TipThis investigation builds genuine understanding rather than rote rule-following. Let your child discover the rule through examples.

One-Step Equations — Review

Solve each equation. Show one step at a time.

x + 7 = 15

3x = −18

x/4 = −3

x − 9 = −2

TipEven simple equations should be solved with clear steps — building good habits here prevents mistakes in harder problems.

Two-Step Equations — Set A

Solve each equation. Show all steps.

2x + 5 = 17

3x − 4 = 11

−2x + 7 = −3

x/3 + 4 = 7

TipFor two-step equations, undo addition/subtraction first, then multiplication/division.

Two-Step Equations — Set B

Solve and check by substituting back.

5(x + 2) = 25

4(2x − 1) = 28

(x + 3)/2 = 7

3(x − 5) + 2 = 11

TipAlways verify by substituting the answer into the original equation.

Equations from Word Problems

Define a variable, write an equation, then solve.

The sum of three consecutive integers is 42. Find the integers.

A rectangle has length 5 m more than its width. The perimeter is 34 m. Find the dimensions.

Two friends are saving for a $480 console. Alex has saved $120 and saves $30/week. Sam has saved $60 and saves $45/week. After how many weeks will they have the same amount?

TipThe hardest part of word problems is setting up the equation. Encourage underlining key information and assigning a variable first.

Solving Inequalities — Set A

Solve each inequality. Write the solution and show on a number line.

x + 3 < 8

2x − 1 ≥ 7

−3x > 12

x/4 ≤ −2

TipFor a number line: open circle (○) means the endpoint is NOT included (strict inequality). Filled circle (●) means it IS included.

Solving Inequalities — Set B

Solve and describe the solution set in words.

3(x + 2) ≤ 15. Solve and describe the solution set.

−2(x − 3) > 10. Solve (remember to flip the sign!) and describe.

5 − 3x < −4. Solve and describe.

TipDescribing the solution in words connects the algebraic solution to its meaning.

Which Inequality Symbol?

Sort each phrase to the correct inequality symbol.

At most $50

More than 100 km

Fewer than 20 students

At least 18 years old

Below 0°C

No more than 2 kg

< (less than)

≤ (less than or equal)

> (greater than)

≥ (greater than or equal)

Check Solutions to Inequalities

Does the given value satisfy the inequality?

x < 5: does x = 4.9 work?

Yes

x ≥ 3: does x = 3 work?

Yes

2x + 1 > 7: does x = 3 work?

Yes

−x ≤ 2: does x = −3 work?

Yes

Equations with Fractions

Solve each equation involving fractions. Show all steps.

x/3 + x/4 = 7

(2x − 1)/5 = 3

x/2 − x/3 = 1

TipMultiply both sides by the LCD (lowest common denominator) to clear fractions — this often makes the problem much easier.

Compound Inequalities

Solve each compound inequality. Show on a number line.

Solve: 2 < x + 3 ≤ 8. Show the solution on a number line.

Solve: −1 ≤ 2x − 3 < 5. Show the solution on a number line.

TipA compound inequality has two conditions at once. Both must be satisfied simultaneously.

Speed, Distance, Time Equations

Use the formula d = st (distance = speed × time). Show all working.

A train travels 360 km in 3 hours. Write and solve an equation to find the speed.

Two cars travel towards each other: Car A at 80 km/h and Car B at 100 km/h. They start 360 km apart. After how many hours do they meet?

TipSpeed, distance, time problems are among the most common real-world applications of linear equations.

Error Analysis — Equations

Find and fix the errors.

Student solves 3x + 5 = 14: '3x = 14 + 5 = 19, x = 19/3.' What is wrong? Find the correct answer.

Student solves −2x > 8: 'x > −4.' What is wrong? Find the correct answer.

TipCommon equation errors: not applying operations to both sides, errors distributing negatives, not flipping inequality sign.

Mixed Equations Review

Solve each equation. Show all steps and verify.

3(2x − 5) + 4 = 2(x + 7)

(x + 3)/4 = (2x − 1)/6

5 − (2x − 3) = 3(x + 1) + 2

TipThese problems mix all equation types covered in this worksheet.

Inequalities in Real Life

Set up and solve an inequality for each situation.

A cinema can hold at most 250 people. 78 people are already seated. Write and solve an inequality to find how many more people can enter.

You need to score at least 75 on your next test to pass a course. Your current average from 3 tests is 70. The next test is worth 20% of the total. What score do you need?

TipReal-world inequalities often involve 'at least', 'at most', 'no more than', or 'no fewer than'.

Reflection

Summarise what you have learned about equations and inequalities.

What is the key difference between solving an equation and solving an inequality?

Give an example where the inequality sign must be flipped. Explain why.

Design your own two-step equation word problem and solve it.

Draw here

TipWritten reflection consolidates understanding and prepares for revision.

What to do next

Measurement

Perimeter & Area of Composite Shapes

Solve problems involving perimeter and area of composite shapes made from rectangles, triangles and semicircles