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.
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 examplegraphing an inequality with a closed dot
These numbers are not on your problems. The steps are. Follow them with your own numbers.
A delivery van is safe when its load is at most 900 pounds. Graph the loads that are safe.
- Write the inequality: let L = the pounds loaded, so L ≤ 900.
- Mark the boundary point at 900 with a CLOSED dot, because 900 is allowed.
- Solutions are 900 and everything smaller, so draw the arrow pointing left.
- Test a point under the arrow: 875 is allowed, which confirms the direction.
Answer: L ≤ 900: a closed dot at 900 with an 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.
A student passes the unit when the score is at least 70 points. Graph the passing scores.
- Write the inequality: let s = , so s 70.
- Mark the boundary at 70. Open or closed dot? Why?
- Which way does the arrow point? Now draw the number line.
- Test a point under your arrow: is a solution?
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 outA 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 — for x < 9 students sometimes shade right instead of left. Always check the symbol twice: once for the circle (does it have an 'or equal to' line?) and once for the shading direction (does the symbol point toward larger or smaller numbers?).
Lesson 8.5Understand Inequalities and Their Solutions
Version ASupported practice
Learning target I can graph the solutions of an inequality on a number line.
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1Write the name of the correct group on each line.
Sort by circle type: open or closed.
- x > 4
- x < 12
- n > 0
- x ≥ 7
- x ≤ 3
- n ≥ 15
Hint < and > use OPEN circles because the boundary is not included.
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2Circle 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
Hint Every inequality graph asks two questions: is 5 itself a solution (circle type), and which side holds the solutions (shading)?
Show your work -
3Circle 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
Hint Check the symbol: ≥ has the 'or equal' bar. What does that bar mean for the circle at 3?
Show your work -
4Circle 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
Hint Two decisions: does x = 8 make x > 8 true, and are the solutions bigger or smaller than 8?
Show your work -
5Circle 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
Hint Ask two things: does 10 itself satisfy x ≤ 10, and do solutions live above or below 10?
Show your work
-
6Find the mistake. Explain it, then show the correct work.
Spot the Common Mistake
- 1ReadGraph n ≥ 8 on the number line.
- 2Circle choiceOpen circle at 8
- 3Shade directionShade right
- 4Final graphOpen circle at 8, shaded right
Which step has the mistake? Explain what went wrong, then show the correct work.
Hint Look at the symbol: does it have a small line under it, like ≤ or ≥?
Correct workCorrect answer:
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.
Lesson 8.5Understand Inequalities and Their Solutions
Version BCore practice
Learning target I can graph the solutions of an inequality on a number line.
-
1Write the letter of the matching item on each line.
Match each inequality to its graph description
- Item 1
- Item 2
- Item 3
- Item 4
- A
- B
- C
- D
-
2Mark and label each value on the number line.
Read the graph description and write the inequality it represents. Then graph it.
Show your thinking -
3Complete the table. Show how you found each value.
Complete the table comparing how different symbols affect the graph.
Inequality Circle Type Shade Direction Is x = boundary a solution? x > 6 x ≥ 6 x ≤ 6 Show your work
-
4Circle the letter of the best answer. Show how you know.
A number line shows a closed circle at 8 and is shaded to the left. Which inequality does this represent?
- Ax ≤ 8
- Bx < 8
- Cx ≥ 8
- Dx > 8
Show your work
-
5Find the mistake. Explain it, then show the correct work.
Spot the Common Mistake
- 1ReadThe case file needs fewer than 9 clues, so c < 9. Graph it.
- 2Circle choiceOpen circle at 9
- 3Shade directionShade right
- 4Final graphOpen circle at 9, shaded right
Which step has the mistake? Explain what went wrong, then show the correct work.
Correct workCorrect answer:
Explain your thinkingHow do you decide between an open circle and a closed circle when graphing?
Lesson 8.5Understand Inequalities and Their Solutions
ChallengeExtension
Learning target I can graph the solutions of an inequality on a number line.
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1Mark and label each value on the number line.
Graph the compound constraint: the suspect's age a is at least 18 AND at most 35. Graph a ≥ 18 and a ≤ 35 on the same number line. What range of ages is possible?
Show your thinking
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2Find the mistake. Explain it, then show the correct work.
Find the Graphing Error
- 1Inequalityx > 6
- 2Circle typeClosed circle at 6
- 3Shade directionShade right
- 4GraphClosed circle at 6, shaded right
Which step has the mistake? Explain what went wrong, then show the correct work.
Correct workCorrect answer: -
3Answer in complete sentences.
A theme park has two rides. Ride A requires height h > 48 inches. Ride B requires height h ≥ 48 inches. A child is exactly 48 inches tall. Which ride can they go on? Explain using number line graphs for both inequalities.
Write your answer