043i - Solving Inequalities Algebraically

Lesson 43 — Solving One-Variable Inequalities Algebraically

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

Learning Intentions

• To solve one-variable linear inequalities using inverse operations.
• To understand and apply the rule that multiplying or dividing by a negative number reverses an inequality.

Success Criteria

I can:

1. Solve one-step and two-step inequalities.
2. Explain, using a numerical example, why multiplying or dividing by a negative number reverses the inequality sign.
3. Solve inequalities with the pronumeral on both sides.
4. Express a solution set correctly using inequality notation.

Warmup

(5 minutes — true or false, mini whiteboards)

Test each value against the inequality .

1. Does satisfy it?
2. Does satisfy it?
3. Does satisfy it?
4. What is the largest whole number that satisfies it?

Answers: 1. Yes, ; 2. No, is false; 3. No; 4. .

Discussion: Point out that unlike an equation, many values work — the solution is a set, not a single number.

Activities

Activity 1 — One and Two-step Inequalities (10 min)

I do. Solve .

I do — a two-step example. Solve .

Rule so far: the same inverse-operation steps as equations apply — add, subtract, multiply, divide both sides — as long as we are not multiplying or dividing by a negative. That case is coming next.

We do:

(Answers: 1. ; 2. .)

You do:

(Answers: 1. ; 2. ; 3. ; 4. .)

Activity 2 — The Negative-multiplier Rule (10 min)

I do — building the rule from a true statement. Start with a true inequality: .

Multiply both sides by : this gives and . Is ? No — that’s false. But is true.

Rule: multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

I do — applying it. Solve .

Check the rule holds: try : ✓. Try (outside the solution set): , not ✓ correctly excluded.

We do:

(Answers: 1. ; 2. .)

You do:

(Answers: 1. ; 2. ; 3. ; 4. .)

Activity 3 — Applied Inquiry: Budgets and Constraints (14 min)

Pairs. Form an inequality for each context, solve it, and interpret the solution in words.

Problem 1. A school trip costs 12$40$400$. How many students can attend?

Problem 2. Mia has 50$8$10$ in reserve. How many books can she buy?

Problem 3. A lift has a safe working load of kg. Each person weighs on average kg, and there is kg of equipment already inside. How many people can safely enter?

Socratic scaffolding for Problem 3:

Prompt	Purpose
Understand: what quantity must stay within a limit?	Total load must not exceed kg.
What varies, and what is fixed?	Number of people varies; equipment mass kg is fixed.
Devise a plan: form an inequality	.
Carry out the plan	, so .
Look back — is a fractional or negative answer sensible here?	No — must be a non-negative whole number, so the practical maximum is people.
Look back — check the boundary	exactly ✓ at the limit; ✗ confirms is unsafe.

Answers: Problem 1 — , so up to students. Problem 2 — , so up to books. Problem 3 — , so up to people.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Solve .
2. Solve .
3. Solve .
4. A vending machine budget allows at most 45$6$3n$.
5. Reasoning. Explain why becomes , not , when solved.

Answers: 1. ; 2. ; 3. ; 4. ; 5. Dividing both sides by (a negative number) reverses the inequality sign, since dividing by a negative reverses order on the number line — confirmed by testing a value either side of .

Common Misconceptions

Misconception	How to pre-empt it
Forgetting to reverse the inequality sign when multiplying or dividing by a negative.	Insist students state “reverse — negative” as an annotation every time it applies.
Reversing the sign when only adding or subtracting a negative number.	Contrast (no reversal, just add ) with (reversal, dividing by ) side by side.
Treating the solution to an inequality as a single value.	Always require students to test at least two values from their claimed solution set.
Writing as incorrectly, reversing the meaning.	Practise reading both directions aloud: ” is less than ” versus ” is less than .”
In context problems, ignoring that the variable must be a non-negative whole number.	Require a “look back” sentence stating any real-world restriction on the variable.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Solve .

Answer

So .

E2 (AMC Junior style). How many positive whole numbers satisfy ?

Answer

. Positive whole numbers: — 5 numbers.

E3 (Challenge). Solve (a compound inequality — apply the same operation to all three parts).

Answer

E4 (Challenge). The inequality has solution . Find .

Answer

Since , we need , so . Check: ✓

Homework

1. Solve: (a) (b) (c) (d) .
2. Solve: (a) (b) (c) (d) .
3. Solve .
4. A car park charges 5$3$26h$.
5. Reasoning. A student solves and writes without reversing the sign. Explain the error and give the correct solution, showing a check with a test value.
6. Challenge. Solve and list all integer solutions.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) (d) . Q3 — . Q4 — . Q5 — dividing by requires reversing the sign; correct solution is ; check : ✓ satisfies but not the student’s , confirming the error. Q6 — ; integers: .