GRADE 7 · ALGEBRA

Equations and Inequalities

Learn how to find unknown values, decide whether a statement is true, and describe a whole set of possible answers. This lesson builds on simplifying expressions and solving two-step equations.

Estimated learning time: 25–35 minutes · Beginner to intermediate

Learning goals

Tell the difference between an expression, an equation and an inequality.
Solve multi-step equations and check a solution by substitution.
Solve simple inequalities and graph their solutions on a number line.
Translate short real-life statements into algebra.

1. Expressions, equations and inequalities

An expression is a mathematical phrase such as 3x + 5. It has no equals or comparison sign. An equation says that two expressions have the same value, using =. An inequality compares values that may not be equal.

Symbol	Meaning	Example
=	is equal to	x + 2 = 9
<	is less than	x < 9
>	is greater than	x > 9
≤	is less than or equal to	x ≤ 9
≥	is greater than or equal to	x ≥ 9

In an equation, a solution is a value that makes the two sides equal. In an inequality, a solution is any value that makes the comparison true. An inequality can therefore have many solutions.

2. Keep an equation balanced

Think of an equation as a balanced scale. Whatever operation you perform on one side, perform the same operation on the other side. Use inverse operations to isolate the variable:

Addition and subtraction undo each other.
Multiplication and division undo each other.

Example A: solve 3x + 5 = 20

3x + 5 = 20

Subtract 5 from both sides: 3x = 15.

Divide both sides by 3: x = 5.

Check: 3(5) + 5 = 15 + 5 = 20. The solution is correct.

Example B: variable terms on both sides

5x + 4 = 2x + 19

Subtract 2x from both sides: 3x + 4 = 19.

Subtract 4: 3x = 15. Divide by 3: x = 5.

Check: left side = 5(5) + 4 = 29; right side = 2(5) + 19 = 29.

Example C: use the distributive property first

2(x + 3) = 18

Expand: 2x + 6 = 18. Subtract 6: 2x = 12. Divide by 2: x = 6.

Helpful order: expand brackets if needed, collect variable terms on one side, collect number terms on the other, then divide or multiply to isolate the variable. Keep both sides balanced at every step.

3. Solve inequalities

You can usually solve an inequality in the same way as an equation. There is one important rule: when you multiply or divide both sides by a negative number, reverse the inequality sign.

Example D: add or subtract

x + 4 > 10

Subtract 4 from both sides: x > 6.

Any number greater than 6 makes the statement true, such as 7, 8 or 10.

Example E: divide by a negative number

−2x < 8

Divide both sides by −2 and reverse < to >: x > −4.

Check with x = 0: −2(0) < 8 becomes 0 < 8, which is true.

Remember: adding or subtracting the same number does not reverse the sign. Multiplying or dividing by a positive number does not reverse it either. Only multiplying or dividing by a negative number reverses the inequality sign.

4. Show solutions on a number line

A number-line graph makes an inequality easier to see.

Use an open circle for < or > because the endpoint is not included.
Use a filled circle for ≤ or ≥ because the endpoint is included.
Shade or draw an arrow toward the values that satisfy the inequality.

Inequality	Endpoint	Direction
x > 3	Open at 3	Right
x ≥ 3	Filled at 3	Right
x < −2	Open at −2	Left
x ≤ −2	Filled at −2	Left

For example, x ≥ 3 includes 3 and every number greater than 3. The endpoint is part of the solution set.

5. Turn words into algebra

Read the statement carefully, identify the unknown, and decide whether it describes an exact value or a limit.

Example F: an equation word problem

Three identical notebooks and a £2 pen cost £17 altogether. Let n be the price of one notebook.

3n + 2 = 17

Subtract 2: 3n = 15. Divide by 3: n = £5.

Example G: an inequality word problem

A club needs at least 24 members. It already has 15. Let m be the number of additional members needed.

15 + m ≥ 24

Subtract 15: m ≥ 9. The club needs at least 9 more members.

Language clues: “is”, “equals” or “the same as” often indicate an equation. “More than”, “fewer than”, “at least” and “no more than” indicate inequalities. Translate the meaning, not just individual words.

6. Common mistakes to avoid

Changing only one side of an equation. Apply the same operation to both sides.
Combining unlike terms, such as 3x + 4 into 7x. The x-term and constant are different types.
Forgetting to reverse the inequality sign when dividing or multiplying by a negative number.
Using an open circle for ≤ or ≥. Those symbols include the endpoint.
Giving one answer when an inequality represents a range of values.
Skipping the check. Substitute your answer into the original equation or test a value in the inequality.

7. Practice questions

Try each question before reading its solution.

Solve: 4x + 7 = 31.
Show solution
Subtract 7: 4x = 24. Divide by 4: x = 6.
Solve: 6x − 5 = 3x + 10.
Show solution
Subtract 3x: 3x − 5 = 10. Add 5: 3x = 15. Therefore x = 5.
Solve: 3(x − 2) = 21.
Show solution
Expand: 3x − 6 = 21. Add 6: 3x = 27. Therefore x = 9.
Solve: x − 8 ≤ 4.
Show solution
Add 8 to both sides: x ≤ 12.
Solve: 5x > −20.
Show solution
Divide by positive 5; the sign stays the same: x > −4.
Solve: −3x ≥ 12.
Show solution
Divide by −3 and reverse ≥ to ≤: x ≤ −4.
A taxi charges a £4 starting fee plus £2 per mile. The total is £18. How many miles were travelled?
Show solution
Let d be miles: 4 + 2d = 18. Subtract 4: 2d = 14. Divide by 2: d = 7 miles.
You have £30 and want to buy notebooks costing £4 each. What is the greatest whole number n you can buy without spending more than £30?
Show solution
4n ≤ 30, so n ≤ 7.5. Since n must be a whole number, the greatest possible number is 7 notebooks.

9. Printable worksheet

Use these questions for independent practice. Show each step and check your answers after attempting the worksheet.