046i - Problem Solving with Equations and Inequalities

Lesson 46 — Problem Solving and Consolidation: Linear Equations and Inequalities

Strand: Algebra | Descriptor: AC9M8A02 | Duration: 45 minutes

Learning Intentions

• To consolidate algebraic and graphical methods for solving linear equations and inequalities.
• To apply these methods to solve multi-step, real-world problems.

Success Criteria

I can:

1. Choose an efficient algebraic or graphical method to solve an equation or inequality.
2. Form an equation or inequality from a worded context and solve it.
3. Verify solutions and interpret them sensibly in context, including inequality endpoints.
4. Communicate my reasoning clearly, following Polya’s problem-solving cycle.

Warmup

(6 minutes — sorting, pairs)

Sort these into “equation” or “inequality,” then classify each by number of steps needed, without solving.

Answers: Equations — (a), (c), (e). Inequalities — (b), (d), (f). (d) will require a sign reversal; (f) requires clearing a fraction across the whole side.

Activities

Activity 1 — Mixed Practice by Type (12 min)

Graded set. Students identify the type and any special feature (fraction, negative coefficient, both sides) before solving.

Set A — equations

Set B — inequalities

(Answers: 1. ; 2. ; 3. ; 4. ; 5. ; 6. ; 7. ; 8. .)

Efficiency discussion: for Q3, ask whether adding to both sides or subtracting is more efficient, and why avoiding a negative coefficient on the variable side is generally preferable.

Activity 2 — Applied Problems (14 min)

Pairs. Full protocol: define the variable, form the equation/inequality, solve, verify, and answer in a sentence.

Problem 1. A gym membership costs 25$15$160$. How many weeks has the member attended?

Problem 2. A delivery van can carry at most kg. It already holds kg, and each box weighs kg. How many more boxes can it carry?

Problem 3. Two water tanks are being filled. Tank A starts with L and fills at L/min. Tank B starts with L and fills at L/min. After how many minutes do the tanks hold equal amounts of water?

Problem 4. A school wants to hire buses for an excursion. Each bus seats students, and the school has students attending plus staff. How many buses are needed at minimum?

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand the problem — what total needs to be seated?	people.
Devise a plan — what inequality models “enough seats”?	, where is the number of buses.
Carry out the plan
Look back — can you order buses?	No — round up to the next whole bus, since a partial bus still costs a full bus and doesn’t fully seat everyone.
Look back — verify	buses seat — not enough. buses seat — enough. So buses are needed.

Answers: Problem 1 — weeks. Problem 2 — up to more boxes. Problem 3 — minutes. Problem 4 — buses.

Activity 3 — Inquiry: Build Your Own Constraint (7 min)

Pairs.

Design a real-world scenario (like the phone-plan or bus problems above) that requires an inequality, not an equation, to solve. Your scenario must include:

• a fixed starting amount or fee,
• a rate that applies per unit,
• a limit that must not be exceeded or must be met.

Write the inequality, solve it, and swap with another pair to check each other’s solution by substitution.

Teacher prompt if pairs stall: “What real situations have a maximum or minimum rather than an exact target? Think about budgets, capacities, weight limits, or minimum scores.”

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Solve .
2. Solve .
3. A taxi charges 4$2.50$29d$ they can afford, and solve it.
4. Verify whether satisfies .
5. Reasoning. Explain why the answer to Problem 4 (the bus problem) required rounding up rather than rounding to the nearest whole number.

Answers: 1. ; 2. ; 3. km; 4. LHS, and ✓ yes; 5. Rounding to the nearest whole number could give a bus count too small to seat everyone; the constraint is a minimum requirement (enough seats), so any leftover people force an extra whole bus, regardless of how small the remainder is.

Common Misconceptions

Misconception	How to pre-empt it
Rounding a context answer to the nearest whole number instead of considering what the situation requires.	Always ask “does rounding up or down make sense here?” as a explicit look-back step.
Choosing a solving method before checking whether the problem is an equation or an inequality.	Require classification (Warmup, Activity 1) before any solving begins.
Forgetting to reverse an inequality sign when the context naturally produces a negative coefficient.	Flag “rate is being subtracted” scenarios (e.g. spending down a budget) as a trigger to check for sign reversal.
Leaving the answer as an inequality in without re-reading the original question’s units or context.	Enforce the “answer in a sentence” step for every applied problem.
Assuming a boundary value is automatically included without testing it against the context.	Model testing the boundary explicitly, as in Problem 4’s look-back step.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A number satisfies and . How many integer values of satisfy both?

Answer

; . Combined: . Integers: — 5 values.

E2 (AMC Junior style). Two plans, and , represent costs for units. For what values of is the first plan cheaper?

Answer

. The first plan is cheaper once exceeds units.

E3 (Challenge). A rectangle’s length is cm more than twice its width, and its perimeter is at most cm. Find the greatest possible width.

Answer

Let width , length . . Greatest width is cm.

E4 (Challenge). Find all whole numbers for which and could both represent the same length in centimetres and satisfy .

Answer

Equal: . Check constraint: ✓. So works, giving a length of cm.

Homework

1. Solve: (a) (b) (c) (d) .
2. A parking garage charges 6$2.50$21h$.
3. Two candles burn down at different rates. Candle A is cm tall and burns cm/hour. Candle B is cm tall and burns cm/hour. After how many hours are they the same height?
4. A lift can safely carry kg. It already holds kg of equipment, and each person weighs about kg. What is the maximum number of people who can safely enter?
5. Verify whether satisfies both and .
6. Reasoning. Explain why forming the correct equation or inequality (Understand and Devise a plan) is usually harder than the algebra itself (Carry out the plan) in worded problems. Give an example from this lesson.
7. Challenge. A school fun run charges 5$0.20$25806$ minutes per lap. Determine whether the runner can reach the fundraising target within the time limit, showing full working.

Answers: Q1 — (a) (b) (c) (d) . Q2 — hours. Q3 — hours. Q4 — (rounding down), so people. Q5 — LHS ✓; LHS ✓ — satisfies both. Q6 — forming the equation requires translating context and deciding relationships (e.g. which quantity is fixed, which rate applies, what the limit means), which involves interpretation and judgement, whereas solving is a mechanical, rule-based process once the equation is correctly set up; e.g. Problem 4’s bus scenario required recognising the need to round up, which is a modelling decision, not an algebraic one. Q7 — laps possible: , so at most laps; sponsorship raised at laps: 2.60$5=$7.60$25$ — the runner cannot reach the target within the time limit.