Graphing Inequalities on a Number Line

Show your work for each problem.

1.
What type of circle should you use for $x < 7$ ?
1. a)Closed circle (included)
2. b)Open circle (not included)
3. c)Either type works
4. d)No circle needed
Answer: Open circle (not included)
$<$ is a strict inequality (7 is not included), so we use an open circle to show that 7 is NOT part of the solution.
2.
For $x \geq -3$ , which direction should you shade?
1. a)Right (toward larger numbers)
2. b)Left (toward smaller numbers)
3. c)Both directions
4. d)No shading needed
Answer: Right (toward larger numbers)
$\geq$ means 'greater than or equal to', so we shade to the RIGHT, toward larger numbers. All values greater than or equal to $-3$ are solutions.
3.
For $x \leq 5$ , at which number do you place the circle?
Answer: 5
The boundary point is 5. We place a closed circle at 5 (because $\leq$ includes equal to) and shade left.
4.
A graph shows a closed circle at $-2$ with shading to the right. Which inequality does it show?
1. a) $x > -2$
2. b) $x < -2$
3. c) $x \leq -2$
4. d) $x \geq -2$
Answer: $x \geq -2$
Closed circle = boundary included ( $\geq$ or $\leq$ ). Shading right = greater than. Together: $x \geq -2$ .
5.
What is the largest integer that is a solution to $x < 4$ ?
Answer: 3
4 is NOT a solution, because $x < 4$ means $x$ must be strictly LESS than 4 (that is why the circle at 4 is open). The largest integer less than 4 is 3.
6.
Which of these numbers is NOT a solution to $x > -5$ ?
1. a) $-7$
2. b) $10$
3. c) $-4$
4. d) $0$
Answer: $-7$
$-7$ is NOT a solution because $-7 < -5$ (it is further left on the number line). The other numbers ( $-4$ , $0$ , $10$ ) are all greater than $-5$ .
7.
Write the inequality: open circle at 3, shading to the left. Use x and the correct symbol.
Answer: x < 3
Open circle = strict inequality (< or >). Shading left = less than. The boundary is 3. So the inequality is $x < 3$ .
8.
Graph the inequality $x \leq -1$ step by step.
Answer: Closed circle at $-1$ , shade left
1. What is the boundary point? -1
2. Does $\leq$ include the boundary? yes
3. What type of circle: open or closed? closed
4. Which direction do you shade: left or right? left
9.
The inequality $5 > x$ means the same as which of the following?
1. a) $x < 5$
2. b) $x > 5$
3. c) $x \geq 5$
4. d) $x \leq 5$
Answer: $x < 5$
$5 > x$ means '5 is greater than x', which is the same as 'x is less than 5', or $x < 5$ . When you swap the two sides, the symbol turns around too.
10.
Solve $x - 2 \geq 5$ and describe how to graph the solution.
Answer: $x ≥ 7$ : closed circle at 7, shade right
1. Add 2 to both sides: $x - 2 + 2 \geq 5 + 2$ , so $x \geq \square$ . What number goes in the box? 7
2. What type of circle goes at 7? closed
3. Which direction do you shade? right
11.
The graph shows an open circle at $-4$ with shading to the left. Write the inequality using x.
Answer: x < -4
An open circle at $-4$ means the boundary is NOT included (use $<$ or $>$ ). Shading left means less than. So: $x < -4$ .
12.
Which value is a solution to $x \leq -6$ ?
1. a) $-4$
2. b) $-10$
3. c) $0$
4. d) $-5$
Answer: $-10$
$-10$ is a solution because $-10 < -6$ (it is further left on the number line). The other values ( $-5$ , $-4$ , $0$ ) are all greater than $-6$ .
13.
A roller coaster requires riders to be at least 140 cm tall. Write and graph this as an inequality, using $h$ for height.
Answer: $h ≥ 140$ : closed circle at 140, shade right
1. Which symbol means 'at least'? ≥
2. Write the inequality using h: h ≥ 140
3. Is a rider who is exactly 140 cm tall allowed? yes
4. What type of circle goes at 140? closed
14.
Solve $2x < 10$ . What is the boundary point of the graph?
Answer: 5
$2x < 10$ → divide both sides by 2 → $x < 5$ . The boundary point is 5. Graph it with an open circle at 5 and shade left.
15.
For food safety, the temperature must stay below 8°C. Write the inequality using $t$ and describe the graph.
Answer: $t < 8$ : open circle at 8, shade left
1. Does 'below 8°C' include exactly 8°C? no
2. Which symbol means 'less than, but not equal'? <
3. Write the inequality: t < 8
4. What type of circle goes at 8? open
16.
A number line shows an open circle at 0 with shading to the right. Which statement is TRUE?
1. a)Zero is a solution
2. b)Negative numbers are solutions
3. c)All real numbers are solutions
4. d)All positive numbers are solutions
Answer: All positive numbers are solutions
This graph shows $x > 0$ . The solutions are all positive numbers (0.5, 1, 2, 3 and so on). Zero is NOT a solution (open circle), and negative numbers are not shaded.