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

Lesson 8.68.6 Small Group · Group 2

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can solve an inequality and graph its solution set on a number line, explain what each part stands for, and build one for a situation I have not seen before.

Why This Matters This is your challenge small group for Lesson 8.6. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
SolveSpanish: Resolver To find the number that makes it true. x + 3 > 10 → subtract 3 → x > 7 — any number greater than 7 works
GraphSpanish: Graficar To show answers on a number line with circles and shading. x > 7: open circle at 7, shade right — shows all solutions at a glance
SolutionSpanish: Solución A number that makes the equation or inequality true. x = 8 is a solution to x > 7 because 8 > 7 is true; x = 5 is not because 5 > 7 is false
SubstituteSpanish: Sustituir To put a number in place of a letter. Substitute x = 5 into x + 3 > 10: 5 + 3 = 8, and 8 > 10 is false, so x = 5 is not a solution
Solution setSpanish: Conjunto solución All the numbers that make the inequality true. For x > 4, the solution set is every number greater than 4, such as 5, 6, 7, ...
Inverse operationSpanish: Operación inversa Two math actions that undo each other, like × and ÷. To solve x + 3 > 7, use the inverse of adding 3: subtract 3 to get x > 4.

How it worksWorked example

These numbers are not on your problems. The steps are. Follow them with your own numbers.

Tickets cost $4 each. Mia can spend at most $28. Write and solve an inequality for the number of tickets t. Then graph it.

  1. Write the inequality from the story: 4t ≤ 28
  2. Divide both sides by 4: t ≤ 28 ÷ 4 → t ≤ 7
  3. Graph the solution: closed circle at 7 → shade left
  4. Check the boundary in the story: 4 · 7 = 28 → 28 ≤ 28 ✓
  5. t ≤ 7. Mia can buy 7 tickets or fewer.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

Solve x − 5 ≤ 8. Then graph the solution and explain the symbol.

Answer:

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • Solving an inequality is like solving an equation because I still use , but it is different because the answer is instead of .

Word bank SolveGraphSolutionSubstituteSolution setInverse operationinverse operationsmany valuesone valuegraphsolution setnumber line

Watch out!A common mistake

A common mistake in Solve and Graph Inequalities is shading the number line in the wrong direction after solving. For example, a student solves x + 4 > 10 correctly to get x > 6, but then shades LEFT toward smaller numbers instead of RIGHT toward greater numbers. The shading direction always matches the solved symbol, not the original problem: '>' and '≥' shade right (greater numbers), '<' and '≤' shade left (smaller numbers)

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

Lesson 8.68.6 Small Group · Group 2

Group 2Challenge practice

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

Learning target I can solve an inequality and graph its solution set on a number line, explain what each part stands for, and build one for a situation I have not seen before.

Why This Matters This is your challenge small group for Lesson 8.6. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.

  1. 1Complete each step, then write the final answer.
    01234567891011120 pens
    Find the boundary. Then graph it.

    Pens cost $3 each. Jo can spend at most $24. Write and solve an inequality for the number of pens p. Then graph it.

    1. 1Write the inequality from the story.
    2. 2Divide both sides by 3. What is the boundary number?
    3. 3Which circle do we draw? Which way do we shade?
    4. 4Check the boundary. Is 3 × 8 at most 24?
    Final answer:
  2. 2Complete each step, then write the final answer.
    15161718192021222324252627test 24
    Find the boundary. Then graph it.

    Solve x ÷ 3 > 7. Then graph it and explain the symbol.

    1. 1Multiply both sides by 3. What is the boundary number?
    2. 2Which circle do we draw? Which way do we shade?
    3. 3Is 21 a solution? Is 24?
    4. 4Why does the > symbol stay the same after multiplying by 3?
    Final answer:
  3. 3Complete each step, then write the final answer.

    Solve 6x ≥ 45. What is the boundary number?

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

    Solve x ÷ 2.5 < 4. What is the boundary number?

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

    A gym charges $12 per class. Lia wants to spend no more than $96. Write and solve an inequality for the number of classes c. What is the greatest number she can take?

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

    Kim and Lou solve 5x > 30. Kim gets x > 6. Lou gets x > 150. What is the boundary number?

    Show your work
    Final answer:

Explain your thinkingAfter you solve an inequality, how do you test the boundary to choose an open or closed circle? Use 5x > 30.

ChallengeWrite your own problem

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

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