044i - Representing Inequalities Graphically

Lesson 44 — Representing Inequalities Graphically and on a Number Line

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

Learning Intentions

• To represent the solution set of a one-variable inequality on a number line, using open and closed circles correctly.
• To represent a one-variable inequality on the Cartesian plane as a region bounded by a vertical or horizontal line.

Success Criteria

I can:

1. Represent solutions such as or on a number line with the correct circle convention.
2. Translate between inequality notation, a number-line diagram, and a worded description.
3. Represent and as shaded regions on the Cartesian plane, bounded by a vertical or horizontal line.
4. Solve an inequality algebraically and represent the solution both ways.

Warmup

(5 minutes — matching, mini whiteboards)

For each number-line description, write the matching inequality.

1. All values greater than , not including .
2. All values less than or equal to .
3. All values between and , including both ends.
4. All values less than , not including .

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

Activities

Activity 1 — Number-line Representation (10 min)

I do. Represent on a number line.

Draw a number line, mark with an open circle (since itself is not included), and shade/arrow to the right.

I do — the closed-circle case. Represent : mark with a closed (filled) circle, shade to the left.

Rule to state aloud every time: ” and mean open circle — the boundary is not included. and mean closed circle — the boundary is included.”

I do — a compound inequality. Represent : open circle at , closed circle at , shading between.

We do: Draw number lines for:

You do:

4. Write the inequality shown by a number line with an open circle at , shaded to the left. (Describe verbally or sketch.)

(Answer to 4: .)

Activity 2 — Regions on the Cartesian Plane (10 min)

I do. Represent on the Cartesian plane.

Draw the vertical line as a dashed line (boundary not included), and shade the entire region to the left of it — every point in that region has an -coordinate less than .

I do — a closed boundary. Represent : draw the horizontal line as a solid line (boundary included), and shade the region above it.

Connection to the number line: the vertical-line region is exactly the number-line solution , “stretched” up and down through every value of . The boundary convention (dashed/open vs solid/closed) is identical in both representations.

We do: Sketch the region for and for .

You do:

1. Sketch .
2. Sketch .
3. A region is shaded above a solid horizontal line at . Write its inequality.

(Answer to 3: .)

Activity 3 — Applied Inquiry: Constraints in Context (14 min)

Pairs. Represent each constraint as an inequality, then on both a number line and (where meaningful) the Cartesian plane.

Scenario 1. A rollercoaster requires riders to be at least cm tall.

Scenario 2. A speed sign reads “Maximum 60.”

Scenario 3. A lift’s indicator shows it is safe for loads under kg, and the manufacturer wants a poster showing all safe loads (in kg) on a number line, from up to the limit.

Scenario 4 (richer). A café is open only when the temperature (°C) satisfies . Represent this on a number line, and describe in words the two boundary behaviours (why one end is closed and the other open).

Socratic scaffolding for Scenario 4:

Prompt	Purpose
Understand: what does "" mean physically?	The café is open exactly at °C — the boundary counts.
What does "" mean physically?	At exactly °C the café is not open — the boundary is excluded.
Devise a plan for the diagram	Closed circle at , open circle at , shading between.
Carry out the plan	Draw and label the number line accordingly.
Look back	Test (open, included) and (closed, excluded) against the rule to confirm the diagram matches the context.

Answers: Scenario 1 — , closed circle at , shaded right. Scenario 2 — , closed circle at , shaded left. Scenario 3 — , closed circle at , open circle at . Scenario 4 — closed circle at , open circle at , shaded between.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Draw (describe) a number line for .
2. Write the inequality for a number line with a closed circle at and an open circle at , shaded between.
3. Describe the region on the Cartesian plane for : which line is the boundary, is it dashed or solid, and which side is shaded?
4. Solve and represent the solution on a number line.
5. Reasoning. Explain why the boundary of is drawn solid but the boundary of is drawn dashed.

Answers: 1. Closed circle at , shaded left; 2. ; 3. Horizontal line , dashed, shaded above; 4. , closed circle at , shaded left; 5. includes the boundary value itself as part of the solution, so it is drawn solid to show inclusion; excludes it, so it is drawn dashed (or open) to show exclusion.

Common Misconceptions

Misconception	How to pre-empt it
Using an open circle for or a closed circle for .	Chant the rule every time: “line under the symbol means the boundary counts — closed circle.”
Shading the wrong side of a vertical or horizontal boundary line.	Test a point not on the line (e.g. the origin) against the inequality to confirm which side to shade.
Treating the Cartesian-plane region as needing both and conditions when only one variable is restricted.	Emphasise that places no restriction on at all — the region extends infinitely up and down.
Reversing which end of a compound inequality is open/closed.	Read the inequality left to right and annotate each symbol separately before drawing.
Believing a dashed boundary line means “no line is drawn.”	Show that a dashed line still marks the exact boundary — it is a visual convention, not an absence of the line.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). How many integers satisfy ?

Answer

Integers: — 7 integers.

E2 (AMC Junior style). A number line shows the solution to an inequality as a closed circle at shaded to the right, and a separate closed circle at shaded to the left. What compound inequality does this represent, and is it possible for both parts to be true at once for some ?

Answer

— yes, all values from to inclusive satisfy both parts simultaneously.

E3 (Challenge). The region and the region together cover the entire number line with no gaps and no overlap, except possibly at one point. What must be true about and ?

Answer

We need , with one region open and the region boundaries meeting exactly — e.g. (open) and (closed) together cover every real number exactly once.

E4 (Challenge). Solve and simultaneously, and describe the resulting number-line diagram.

Answer

First: . Second: . Combined: — open circle at , closed circle at , shaded between.

Homework

1. Draw (describe in words) a number line for: (a) (b) (c) .
2. Write the inequality shown by each description: (a) closed circle at , shaded right (b) open circle at , shaded left (c) open circle at and closed circle at , shaded between.
3. Describe the Cartesian-plane region for: (a) (b) .
4. Solve and represent the solution on a number line.
5. A car park is open for vehicles under m tall. Write this as an inequality and describe its number-line diagram.
6. Reasoning. Explain, using an example, the difference between representing an inequality on a number line versus on the Cartesian plane. What extra information does the Cartesian-plane version carry?
7. Challenge. Solve and list the integer solutions.

Answers: Q1 — (a) open circle at , shaded right (b) closed circle at , shaded left (c) closed circle at , open circle at , shaded between. Q2 — (a) (b) (c) . Q3 — (a) solid vertical line at , shaded right (b) dashed horizontal line at , shaded below. Q4 — , open circle at , shaded right. Q5 — , open circle at , shaded left. Q6 — a number-line diagram shows only the solution values for one variable; a Cartesian-plane region shows that the same restriction applies for every value of the other variable, so it carries the extra information that the condition holds across an entire infinite strip of the plane. Q7 — ; integers: .