Section A — Recall Questions 1–5
1.
State the meaning of the symbol $\geq$ in words.
Answer
Greater than or equal to.
2.
What type of circle is used on a number line to represent a strict inequality?
Answer
An open circle.
Remember: Open circle = strict inequality ($<$ or $>$) — boundary value not included. Closed circle = inclusive inequality ($\leq$ or $\geq$) — boundary value included.
3.
State the two inequality symbols that require a closed circle on a number line.
Answer
$\leq$ and $\geq$
4.
On a number line, in which direction does the line extend from the circle for the inequality $x > 5$?
Answer
To the right (towards larger values).
5.
What does a closed circle at a value on a number line tell you about that value?
Answer
The value is included in the inequality — it is part of the solution.
Section B — Application Questions 6–10
6.
Write the inequality shown on this number line.
Answer
- Open circle at 3 → 3 is not included → strict inequality
- Line goes to the right → greater than
$x > 3$
7.
Write the inequality shown on this number line.
Answer
- Closed circle at −2 → −2 is included → inclusive inequality
- Line goes to the left → less than or equal to
$x \leq -2$
8.
Write the inequality shown on this number line.
Answer
- Left boundary: closed circle at −2 → −2 is included → $\leq$
- Right boundary: open circle at 2 → 2 is not included → $<$
- Line runs between the two circles → range
$-2 \leq x < 2$
9.
Describe how to represent the inequality $x \geq -3$ on a number line. State: the type of circle used, where to place it, and the direction the line goes.
Answer
- The symbol is $\geq$ (inclusive) → use a closed circle
- Place the closed circle at $-3$
- $\geq$ means greater than or equal to → draw the line to the right
Closed circle at −3, line extending to the right.
10.
Write the inequality shown on this number line.
Answer
- Open circle at 0 → 0 is not included → strict inequality
- Line goes to the right → greater than
$x > 0$
Section C — Challenge Questions 11–15
11.
Write the inequality shown by each of the following number lines.
(a)
(b)
(c)
Answer
(a)
Closed circle at −4 → −4 is included → $\leq$. Line goes left → less than or equal to.
$x \leq -4$
(b)
Open circle at 1 → 1 is not included → $>$. Line goes right → greater than.
$x > 1$
(c)
Closed circle at −1 → −1 is included → $\leq$. Open circle at 3 → 3 is not included → $<$. Line between two circles → range.
$-1 \leq x < 3$
12.
For the inequality $-4 \leq x < 1$:
(a)
What type of circle is used at $x = -4$? Explain why.(b)
What type of circle is used at $x = 1$? Explain why.(c)
Is the value $x = 1$ included in this inequality? Give a reason.Answer
(a) Circle at $x = -4$
Closed circle. The symbol at this boundary is $\leq$, which is inclusive — so −4 is included in the solution.
(b) Circle at $x = 1$
Open circle. The symbol at this boundary is $<$, which is strict — so 1 is not included in the solution.
(c) Is $x = 1$ included?
No. The symbol is $<$ (strictly less than 1), so $x = 1$ is not part of the solution. The open circle at 1 shows this.
Key rule: Always check the symbol at each boundary separately. $\leq$ or $\geq$ → closed circle (included). $<$ or $>$ → open circle (not included).
13.
A student draws a number line to show $x > 2$. They use a closed circle at 2 and shade to the right.
(a)
Identify the mistake in the student’s diagram.(b)
State what the correct circle should be and explain why.Answer
(a) The mistake
The student used a closed circle at 2. A closed circle means 2 is included in the solution, but $x > 2$ means $x$ is strictly greater than 2 — so 2 is not included.
(b) Correct circle
The correct circle is an open circle at 2. The symbol $>$ is a strict inequality, which means the boundary value (2) is not part of the solution. An open circle shows the value is not included.
Common mistake: Using a closed circle for $<$ or $>$. Always ask: is the boundary value included? If not → open circle.
14.
A number line shows a closed circle at $-3$ and an open circle at $0$, with a line connecting them.
(a)
Write the inequality shown.(b)
Is $x = -3$ a solution? Give a reason.(c)
Is $x = 0$ a solution? Give a reason.(d)
Write one value of $x$ that satisfies this inequality.Answer
(a) The inequality
Closed circle at −3 → $\leq$. Open circle at 0 → $<$.
$-3 \leq x < 0$
(b) Is $x = -3$ a solution?
Yes. The closed circle at −3 shows that −3 is included in the inequality (the symbol is $\leq$).
(c) Is $x = 0$ a solution?
No. The open circle at 0 shows that 0 is not included in the inequality (the symbol is $<$).
(d) One value of $x$
Any value between −3 and 0. For example: $x = -2$ (or $x = -1$, etc.)
15.
Answer the following questions about inequalities and number lines.
(a)
A number line has a closed circle at 1 with the line going to the right. Write the inequality this represents.(b)
Write an inequality that would be shown with an open circle at 1 with the line going to the left.(c)
Explain how the two inequalities from (a) and (b) differ in terms of whether $x = 1$ is included.Answer
(a)
Closed circle → inclusive ($\geq$). Line goes right → greater than or equal to.
$x \geq 1$
(b)
Open circle → strict ($<$). Line goes left → less than.
$x < 1$
(c) How they differ
In (a), $x \geq 1$ uses $\geq$, so $x = 1$ is included — shown by the closed circle.
In (b), $x < 1$ uses $<$, so $x = 1$ is not included — shown by the open circle.
The difference is the inequality symbol: $\geq$ includes the boundary value; $<$ does not.
In (b), $x < 1$ uses $<$, so $x = 1$ is not included — shown by the open circle.
The difference is the inequality symbol: $\geq$ includes the boundary value; $<$ does not.
