Grade 6 Mathematics · Unit 8: Equations & InequalitiesStandard 6.AT.9

Lesson 8.5Understand Inequalities and Their Solutions

Start hereWords, worked example, and sentence starters

Name Date Period

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.

WordWhat it meansExample
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.

  1. The boundary number is 6.
  2. < does not include 6, so draw an open dot at 6.
  3. 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?

Answer:

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

Grade 6 Mathematics · Unit 8: Equations & InequalitiesStandard 6.AT.9

Lesson 8.5Understand Inequalities and Their Solutions

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

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?

Part 1Understand the ideamodels and meaning
  1. 1Write the name of the correct group on each line.

    Sort by circle type: open or closed.

    Groups: Open circle (not included) Closed circle (included)
    • x > 4
    • x < 12
    • n > 0
    • x ≥ 7
    • x ≤ 3
    • n ≥ 15

    Hint < and > use OPEN circles because the boundary is not included.

Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    Which graph represents x < 5?

    1. AOpen circle at 5, shaded left
    2. BClosed circle at 5, shaded left
    3. COpen circle at 5, shaded right
    4. 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
  2. 3Circle the letter of the best answer. Show how you know.

    Which graph represents x ≥ 3?

    1. AClosed circle at 3, shaded right
    2. BOpen circle at 3, shaded right
    3. CClosed circle at 3, shaded left
    4. 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
  3. 4Circle the letter of the best answer. Show how you know.

    Which graph represents x > 8?

    1. AOpen circle at 8, shaded right
    2. BClosed circle at 8, shaded right
    3. COpen circle at 8, shaded left
    4. 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
  4. 5Circle the letter of the best answer. Show how you know.

    Which graph represents x ≤ 10?

    1. AClosed circle at 10, shaded left
    2. BOpen circle at 10, shaded left
    3. CClosed circle at 10, shaded right
    4. 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
Part 3Explain and correctreasoning
  1. 6Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake

    1. 1ReadGraph n ≥ 8 on the number line.
    2. 2Circle choiceOpen circle at 8
    3. 3Shade directionShade right
    4. 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 work
    Correct 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.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 8: Equations & InequalitiesStandard 6.AT.9

Lesson 8.5Understand Inequalities and Their Solutions

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

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?

Part 1Understand the ideamodels and meaning
  1. 1Write the letter of the matching item on each line.

    Match each inequality to its graph description

    1. Item 1
    2. Item 2
    3. Item 3
    4. Item 4
    • A
    • B
    • C
    • D
  2. 2Mark and label each value on the number line.

    Read the graph description and write the inequality it represents. Then graph it.

    012345678910
    Show your thinking
  3. 3Complete the table. Show how you found each value.

    Complete the table comparing how different symbols affect the graph.

    InequalityCircle TypeShade DirectionIs x = boundary a solution?
    x > 6
    x ≥ 6
    x ≤ 6
    Show your work
Part 2Apply itreal-world problems
  1. 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?

    1. Ax ≤ 8
    2. Bx < 8
    3. Cx ≥ 8
    4. Dx > 8
    Show your work
Part 3Explain and correctreasoning
  1. 5Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake

    1. 1ReadThe case file needs fewer than 9 clues, so c < 9. Graph it.
    2. 2Circle choiceOpen circle at 9
    3. 3Shade directionShade right
    4. 4Final graphOpen circle at 9, shaded right

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:

Explain your thinkingHow do you decide between an open circle and a closed circle when graphing?

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 8: Equations & InequalitiesStandard 6.AT.9

Lesson 8.5Understand Inequalities and Their Solutions

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

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?

Part 1Understand the ideamodels and meaning
  1. 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?

    012345678910
    Show your thinking
Part 2Explain and correctreasoning
  1. 2Find the mistake. Explain it, then show the correct work.

    Find the Graphing Error

    1. 1Inequalityx > 6
    2. 2Circle typeClosed circle at 6
    3. 3Shade directionShade right
    4. 4GraphClosed circle at 6, shaded right

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:
  2. 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

Explain your thinkingHow do you decide between an open circle and a closed circle when graphing?

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help