Grade 7 · Grade 7: Linear Expressions, Equations and Inequalities · Linear Expressions, Equations and Inequalities
Linear Expressions, Equations and Inequalities
Expand, factorise and rewrite linear expressions, then solve px + q = r, p(x + q) = r and matching inequalities.
Learning objectives
- Add, subtract, expand and factorise linear expressions.
- Explain what a rewritten expression reveals, such as a + 0.05a = 1.05a.
- Solve multi-step problems with rational numbers and check reasonableness.
- Solve and graph inequalities, reversing the sign when multiplying by a negative.
AERO Mathematics alignment
AERO.M7.AE.1
Apply properties of operations to add, subtract, factor and expand linear expressions with rational coefficients.
AERO.M7.AE.2
Understand that rewriting an expression in different forms can show how the quantities in a problem are related.
AERO.M7.AE.3
Solve multi-step real-life problems with positive and negative rational numbers in any form, using tools strategically and assessing reasonableness.
AERO.M7.AE.4
Use variables to represent quantities and construct simple equations of the form px + q = r and p(x + q) = r to solve problems, comparing algebraic and arithmetic approaches.
AERO.M7.AE.5
Solve word problems leading to inequalities of the form px + q > r or px + q < r, graph the solution set and interpret it in context.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
A gym charges 150 QAR joining fee plus 80 QAR per month. Your budget is 800 QAR.
Think about it
How many months can you afford, and how does an inequality answer that better than an equation?
Hint: Write 80m + 150 ≤ 800.
EXPLAIN
Rewriting and solving
Expanding and factorising
The distributive property works both ways: 3(x + 4) = 3x + 12 expands, and 6x + 9 = 3(2x + 3) factorises. Only like terms combine: 5x + 2x = 7x, but 5x + 2 stays as it is.
What a rewritten form reveals
a + 0.05a = 1.05a shows instantly that adding 5% is the same as multiplying by 1.05. Different equivalent forms answer different questions.
Solving in two steps
| Equation | Step 1 | Step 2 | Solution |
|---|---|---|---|
| 3x + 7 = 22 | Subtract 7 | Divide by 3 | x = 5 |
| 5(x − 2) = 35 | Divide by 5 (or expand) | Add 2 | x = 9 |
| −4x = 20 | Divide by −4 | — | x = −5 |
Inequalities
Solve inequalities exactly like equations, with one extra rule: multiplying or dividing both sides by a negative reverses the direction. −3x > 12 becomes x < −4.
Remember
Always check reasonableness. If a budget problem gives m = 8.125 months, the realistic answer is 8 whole months.
INVESTIGATION
Try it yourself
Compare two real subscription plans. Write an expression for each, set them equal to find the break-even point, then write an inequality describing when the first plan is cheaper. Explain your answer in a sentence.
Watch
Solving 2-Step Equations
Math Antics · 9:00
A clean method for px + q = r that scales to harder equations.
Before you watch: Which operation should you undo first in 3x + 5 = 20?
- Solve 5(x − 2) = 35.
- Solve −3x > 12 and explain the sign flip.
Interactive simulation · PhET
Expression Exchange
Use the variables screen to expand and factorise linear expressions.
While you explore
- Which expressions collapse to the same simplest form?
- How does the sim show 3(x + 2) = 3x + 6?
Key vocabulary
- Factorise
- To rewrite a sum as a product.
- Like terms
- Terms with exactly the same variable part.
Practice questions
0/1 correct
Level 1 · Criterion A
Solve 4(x − 3) = 20.
MYP criterion tasks
Level 2 · Criterion C
Solve −5x + 3 > 18 and explain the direction of the inequality sign.
Reflect & track
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