Inequalities and Solution Sets

By the end of this lesson you will be able to:

• Read and write inequalities using , , ,
• Solve linear inequalities, including the sign-flip rule
• Represent solution sets on a number line and as a list
• Apply inequalities to real-life word problems

The Hook: One Answer or Many?

An equation: — only one value works: ✓

But consider:

"You need at least 2 000 UGX to join the school trip."

How many possible amounts satisfy that? Is it one? Ten? Infinitely many?

That is what this lesson answers.

The Four Inequality Symbols: Quick Reference

Strict vs. Inclusive — The Key Distinction

Strict ( and ): boundary excluded

• : 3 itself does not satisfy the condition

Inclusive ( and ): boundary included

• : 3 does satisfy the condition

Open dot () = strict; closed dot () = inclusive:

English Phrases to Inequality Symbols

Check-In 1 — Write as Inequalities

1. "Bag weighs more than 3 kg." ()
2. "At most 6 people in the car." ()
3. "Learner must be at least 12." ()
4. "Price is less than 1 500 UGX." ()
5. "No fewer than 8 items." ()

Solving Inequalities: Balance Method Applies

Why the Balance Method Works Here

Adding/subtracting never changes the inequality direction.

No-flip: add/subtract any number; multiply/divide by positive.

No-Flip Cases: Three Worked Examples

Ex 1: subtract 3:

Ex 2: divide by 2 (positive):

Ex 3:

Why the Sign Flips for Negative Multipliers

Multiply by without flipping: — FALSE.

Flip the sign:

Rule: Multiply or divide by a negative → reverse the sign.

Probe — Find the Error

A student solves :

Step 1:
Step 2: ← the student flipped the sign
Step 3:

What is wrong? Why?

Pause and identify the error before reading on.

Probe Explained: Where the Error Was

Error: Flipped on a subtraction step — flip applies only to dividing/multiplying by a negative.

Correct solution:

Sign-Flip Rule: Two Worked Examples

Check: : ✓ and ✓

Check: : ✓ and ✓

Check-In 2 — Solve; Did You Flip?

(1) — no flip (2) — no flip (3) — flip (4) — flip

From One Answer to a Solution Set

gives — not just .

, , all satisfy it too.

"" is a set — every number left of 5.

How do we write that set clearly?

Solution Set: A New Mathematical Object

Three representations: inequality · number line · listed set.

Three Representations — Worked:

Domain: counting numbers

1. Inequality:
2. Number line: closed dot at 4, arrow left
3. Listed set: (finite)

Three Representations — Worked:

Domain: counting numbers

1. Inequality:
2. Number line: open dot at 3, arrow right
3. Listed set: (infinite)

Solution Set as Subset of the Domain

From sets-01: the universal set is the full domain.

The solution set is a subset of :

Check-In 3 — Solution Sets

1. Three values satisfying .
2. List the solution set of .
3. Number line for — open or closed dot?
4. Solve ; list the solution set.

Word Problems: 5-Step Routine Adapted

Word Problems — Two Worked Examples

P1: Taxi holds at most 14 (): ; set:

P2: Errands pay 500 UGX; need at least 2 000 UGX ():

• ; set:

Unscaffolded Practice — Your Turn

Tuck shop: items cost 500 UGX each. You have 1 800 UGX. How many items can you buy?

Apply all 5 steps independently.

Solution: = items; ; solution set (whole items): . Buy at most 3.

Closer — Three Algebra Tools

• Negative divisor: flip sign; positive: unchanged
• Open dot (, ); closed dot (, )
• Inequalities have solution sets, not single values