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

Lesson 8.58.5 Small Group · Group 2

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can graph the solutions of an inequality on a number line, explain what each part stands for, and build one for a situation I have not seen before.

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 2

Group 2Challenge 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, explain what each part stands for, and build one for a situation I have not seen before.

  1. 1Complete each step, then write the final answer.
    4567891011121314test 7
    Find the boundary. Then shade.

    Graph 10 > x on a number line.

    1. 1Read it with x first. What inequality do you get?
    2. 2Do you draw an open or a closed circle at 10?
    3. 3Which way do you shade?
    4. 4Test x = 7. Is 10 > 7 true?
    Final answer:
  2. 2Complete each step, then write the final answer.
    23456789101112test 9
    Find the boundary. Then shade.

    Graph 7 ≤ y on a number line. Then explain the shading.

    1. 1Read it with y first. What inequality do you get?
    2. 2Do you draw an open or a closed circle at 7?
    3. 3Which way do you shade?
    4. 4Why do you shade right, even though ≤ points left?
    Final answer:
  3. 3Complete each step, then write the final answer.

    Graph 1.5 < x. Which way do you shade from 1.5: left or right?

    Show your work
    Final answer:
  4. 4Complete each step, then write the final answer.
    67891011121314151612
    Closed circle at 12, shaded to the left.

    The number line shows a closed circle at 12, shaded to the left. It shows x ≤ 12. Which symbol goes in the box: 12 ☐ x?

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

    A sign says: "Kids under 6 ride free." Graph the ages a that ride free. Which circle do you draw at 6: open or closed?

    Show your work
    Final answer:
  6. 6Complete each step, then write the final answer.

    Mo graphs 5 > x with an open circle at 5. He shades right because > points right. Which way should he shade: left or right?

    Show your work
    Final answer:

Explain your thinkingMo trusted the direction the symbol points. Why is testing a number a safer way to choose the side to shade?

ChallengeWrite your own problem

Write a new word problem that uses this lesson's idea. Then solve it and show every step.

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