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

Lesson 8.58.5 Small Group · Group 1

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target With my small group, I can graph the solutions of an inequality on a number line — one step at a time, with support.

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 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.

  1. Write the inequality: let L = the pounds loaded, so L ≤ 900.
  2. Mark the boundary point at 900 with a CLOSED dot, because 900 is allowed.
  3. Solutions are 900 and everything smaller, so draw the arrow pointing left.
  4. 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.

  1. Write the inequality: let s = , so s 70.
  2. Mark the boundary at 70. Open or closed dot? Why?
  3. Which way does the arrow point? Now draw the number line.
  4. Test a point under your arrow: is a solution?
Answer:graph of the passing scores

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?).

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

Lesson 8.58.5 Small Group · Group 1

Group 1Supported practice

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

Learning target With my small group, I can graph the solutions of an inequality on a number line — one step at a time, with support.

  1. 1Complete each step, then write the final answer.
    012345678910test 8
    Mark the boundary. Then shade.

    Graph y ≥ 6 on a number line.

    1. 1What is the boundary number?
    2. 2Is 6 included? Do we draw an open or a closed circle?
    3. 3Which way do we shade for "greater than or equal to"?
    4. 4Test y = 8. Is 8 ≥ 6 true?
    Final answer:
  2. 2Complete each step, then write the final answer.
    012345678test 1
    Mark the boundary. Then shade.

    Graph x ≤ 3 on a number line.

    1. 1Is 3 included? Do we draw an open or a closed circle?
    2. 2Which way do we shade for "less than or equal to"?
    3. 3Test x = 1. Is 1 ≤ 3 true?
    Final answer:
  3. 3Complete each step, then write the final answer.

    Graph x > 8. Which circle do you draw at 8: open or closed?

    Show your work

    Hint Is 8 > 8 true? If 8 is not a solution, the circle is empty.

    Final answer:
  4. 4Complete each step, then write the final answer.

    Graph w ≤ 15. Which way do you shade from 15: left or right?

    Show your work

    Hint Pick a number less than 15, like 10. Is 10 ≤ 15 true?

    Final answer:
  5. 5Complete each step, then write the final answer.

    Riders on a ride must be at least 42 inches tall, so h ≥ 42. Which circle do you draw at 42: open or closed?

    Show your work

    Hint Can a rider who is exactly 42 inches tall ride? Then 42 is a solution.

    Final answer:
  6. 6Circle the letter of the best answer. Show how you know.
    0123456784
    Open circle at 4, shaded to the left.

    The number line shows an open circle at 4, shaded to the left. Which inequality does it show?

    1. Ax < 4
    2. Bx ≤ 4
    3. Cx > 4
    4. Dx ≥ 4

    Hint First read the circle: is 4 included? Then read the shading: smaller or greater numbers?

    Show your work

Explain your thinkingHow do the circle and the shading work together to show every solution? Use y ≥ 6 to explain.

Sentence starter The circle at ___ is ___ because ___. · I shade ___ because the solutions are ___.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help