Lesson 8.5Understand Inequalities and Their Solutions
Start hereWords, worked example, and sentence starters
Learning target I can graph the solutions of an inequality on a number line.
Why This Matters Detective Okafor is building a timeline. The suspect was seen at the scene no earlier than 2:00 PM, meaning the arrival time t satisfies t ≥ 2. She needs to plot this on a timeline to share with her team. How should she mark the number line?
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| Number lineSpanish: Recta numérica | A straight line where numbers are placed in order; numbers get smaller to the left and larger to the right. | A horizontal line with marks at 0, 1, 2, 3, ... showing where values fall |
| Open circleSpanish: Círculo abierto | An empty circle showing a number is NOT included. | x > 5: open circle at 5 means 5 itself is NOT a solution, but 5.1, 6, 7, ... are |
| Closed circleSpanish: Círculo cerrado | A filled-in circle showing a number IS included. | used for ≤ and ≥ |
| Solution setSpanish: Conjunto solución | All the numbers that make the inequality true. | every number 8 or above |
| InequalitySpanish: Desigualdad | A math sentence that compares two sides with <, >, ≤, or ≥. | x ≥ 3 is an inequality; it is graphed as a closed circle on 3 with an arrow to the right. |
| Boundary pointSpanish: Punto límite | The number on the number line where the solution set starts. | the 8 in x ≥ 8 |
How it worksWorked example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
Graph x < 6 on a number line.
- The boundary number is 6.
- < does not include 6, so draw an open dot at 6.
- Less than 6 means the arrow points left. Test: 5 < 6 is true.
Answer: open dot at 6, arrow pointing left
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
Graph x < 5. What number is the boundary?
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- My inequality is because the phrase says .
- I drew a dot at because .
- My arrow points because the solutions are .
- The value sits under my arrow, so it .
Word bank inequalitygraphnumber lineboundary pointclosed circleopen circlearrowshadesolution setat leastat mostinclude
Watch out!A common mistake
A common mistake in Graph Inequalities is treating ≥ and ≤ the same as > and < and drawing the wrong circle. For example, students often graph x ≥ 6 with an OPEN circle (as if it were x > 6), which wrongly excludes 6 even though 6 itself makes x ≥ 6 true. A second common mistake is choosing the right circle but shading the wrong direction
Lesson 8.5Understand Inequalities and Their Solutions
Level 0Most support · every hint printed
Learning target I can graph the solutions of an inequality on a number line.
Why This Matters Detective Okafor is building a timeline. The suspect was seen at the scene no earlier than 2:00 PM, meaning the arrival time t satisfies t ≥ 2. She needs to plot this on a timeline to share with her team. How should she mark the number line?
How it worksWorked example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
Graph x < 6 on a number line.
- The boundary number is 6.
- < does not include 6, so draw an open dot at 6.
- Less than 6 means the arrow points left. Test: 5 < 6 is true.
Answer: open dot at 6, arrow pointing left
-
1Circle the letter of the best answer. Show how you know.
Which graph represents x < 5?
- AOpen circle at 5, shaded left
- BClosed circle at 5, shaded left
- COpen circle at 5, shaded right
- DClosed circle at 5, shaded right
Steps to try- Every inequality graph asks two questions: is 5 itself a solution (circle type), and which side holds the solutions (shading)?
- Strict symbols like < exclude the boundary, so check whether 5 makes x < 5 true. Then shade toward test values that work.
- First step: 5 < 5 is false, so 5 is NOT included — that fixes the circle type. Now test x = 4 to find the shading direction.
Show your work -
2Circle the letter of the best answer. Show how you know.
Which graph represents x ≥ 3?
- AClosed circle at 3, shaded right
- BOpen circle at 3, shaded right
- CClosed circle at 3, shaded left
- DOpen circle at 3, shaded left
Steps to try- Check the symbol: ≥ has the 'or equal' bar. What does that bar mean for the circle at 3?
- The bar means 3 IS a solution, so the circle gets filled in. Then a quick test value tells you the shading side.
- First step: 3 ≥ 3 is true, so 3 is included. Now test x = 10 — is 10 ≥ 3 true? Shade toward that side.
Show your work -
3Circle the letter of the best answer. Show how you know.
Which graph represents x > 8?
- AOpen circle at 8, shaded right
- BClosed circle at 8, shaded right
- COpen circle at 8, shaded left
- DClosed circle at 8, shaded left
Steps to try- Two decisions: does x = 8 make x > 8 true, and are the solutions bigger or smaller than 8?
- Symbols without the 'or equal' bar exclude the boundary. Then shade only the side where test numbers make the statement true.
- First step: 8 > 8 is false, so 8 is NOT included — that fixes your circle. Now test x = 9 to pick the shading side.
Show your work -
4Circle the letter of the best answer. Show how you know.
Which graph represents x ≤ 10?
- AClosed circle at 10, shaded left
- BOpen circle at 10, shaded left
- CClosed circle at 10, shaded right
- DOpen circle at 10, shaded right
Steps to try- Ask two things: does 10 itself satisfy x ≤ 10, and do solutions live above or below 10?
- The 'or equal' bar means the boundary is a solution — that decides the circle. A test value decides the shading.
- First step: 10 ≤ 10 is true, so 10 is included. Now test x = 3 — is 3 ≤ 10 true? Shade toward the side that works.
Show your work
Explain your thinkingHow do you decide between an open circle and a closed circle when graphing?
Sentence starter I use a ___ circle when the symbol is ___ because the boundary value ___ part of the solution.