Graphing Inequalities on a Number Line

Show your work for each problem.

1.
What type of circle should be used for ?
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)
Since is a strict inequality (not including 7), we use an open circle to show that 7 is NOT part of the solution.
2.
For , 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)
Since means 'greater than or equal to', we shade to the RIGHT toward larger numbers. All values greater than or equal to are solutions.
3.
For , at which number do you place the circle?
Answer: 5
The boundary point is 5. We place a closed circle at 5 (because includes equal to) and shade left.
4.
A graph shows a closed circle at with shading to the right. Which inequality does this represent?
1. a)
2. b)
3. c)
4. d)
Answer:
Closed circle = boundary included ( or ). Shading right = greater than. Combined: .
5.
Is 4 a solution to ? Answer yes or no.
Answer: no
No, 4 is NOT a solution because means must be strictly LESS than 4, not equal to it. That's why we use an open circle.
6.
Which of these numbers is NOT a solution to ?
1. a)
2. b)
3. c)
4. d)
Answer:
is NOT a solution because (it's further left on the number line). The other numbers (, , ) are all greater than .
7.
Write the inequality: Open circle at 3, shading 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 .
8.
Graph the inequality step by step.
Answer: closed circle at -1, shade left
1. What is the boundary point? -1
2. Does include the boundary? (yes/no) yes
3. What type of circle: open or closed? closed
4. Which direction to shade: left or right? left
9.
The inequality is the same as which of the following?
1. a)
2. b)
3. c)
4. d)
Answer:
means '5 is greater than x', which is the same as 'x is less than 5' or . We flip the inequality when we flip the order.
10.
Solve and describe how to graph it.
Answer: x ≥ 7, closed circle at 7, shade right
1. Add 2 to both sides: . What is ? 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 with shading to the left. Write the inequality using x.
Answer: x < -4
Open circle at means the boundary is NOT included (use or ). Shading left means less than. So: .
12.
Which value is a solution to ?
1. a)
2. b)
3. c)
4. d)
Answer:
is a solution because (it's further left on the number line). The other values (, , ) are all greater than .
13.
A roller coaster requires riders to be at least 140 cm tall. Write and graph this as an inequality using for height.
Answer: h ≥ 140, closed circle at 140, shade right
1. What does 'at least 140 cm' mean? (≤ or ≥) ≥
2. Write the inequality using h: h ≥ 140
3. Is 140 cm tall enough to ride? (yes/no) yes
4. What type of circle at 140? closed
14.
Solve and give the boundary point for graphing.
Answer: 5
→ Divide both sides by 2 → . The boundary point is 5. Graph with an open circle at 5, shading left.
15.
The temperature must stay below 8°C for food safety. Write the inequality using and describe the graph.
Answer: t < 8, open circle at 8, shade left
1. Does 'below 8°C' include exactly 8°C? (yes/no) no
2. What symbol means 'less than but not equal'? <
3. Write the inequality: t < 8
4. What type of circle 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 represents . The solutions are all positive numbers (1, 2, 3, ..., 0.5, etc.). Zero is NOT a solution (open circle), and negative numbers are not shaded.