8.6 Small Group · Group 2

Tier 1 Extension · Non-Routine Synthesis · Misconception Traps · Rigorous CER Proofs

Solve and Graph Inequalities · Solve · Procedural Fluency · Reasoning & Critique

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — How do I solve an inequality and then graph it?

1Solve with inverse operations, then graph the solution set: open circle for < or >, closed for ≤ or ≥ — and you can say why it is true, and where it would stop being true.

You already can do this. So we go up a level, not up a number: find what is always true, show it a second way, and be ready to defend it with a reason instead of an answer.

2Worked Example — Watch me

Solve x + 3 > 53. The inverse of adding 3 is subtracting 3.
I subtract 3 from both sides: x + 3 − 3 > 53 − 3, so x > 50.
The symbol > does not include 50, so I use an open circle at 50.
'Greater than' shades right. Check: x = 60 gives 60 + 3 = 63 > 53. True!

3Second Model — Try it together — then prove it

Solve x − 5 ≤ 8. What is the inverse of subtracting 5?
We add 5 to both sides: x ≤ 8 + 5, so x ≤ 13.
≤ includes 13, so we use a closed circle and shade left.
Substitute x = 10: 10 − 5 = 5 ≤ 8. True?
Now prove it: say why that move had to work at all — not just that it did.
Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."

4Mathematical Word Bank

Solve and Graph Inequalities (Resolver y representar desigualdades) — Isolate the variable, then show the complete solution set on a number line.
Solve (Resolver) — To find the number that makes it true.
Graph (Graficar) — To show answers on a number line with circles and shading.
Solution (Solución) — A number that makes the equation or inequality true.
Substitute (Sustituir) — To put a number in place of a letter.
Solution set (Conjunto solución) — All the numbers that make the inequality true.

5Watch out

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) — check this AFTER you solve, not before. To catch this mistake, substitute one shaded value and one unshaded value back into the original inequality: the shaded value should make it true, and the unshaded value should make it false.

SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 FILL TABLE

Solve each inequality, describe the inverse operation, and describe the graph.

Inequality Inverse Operation Solution Graph Description
x + 11 ≥ 20
x − 3 < 7
x + 8 ≤ 20

✏️ Scratchpad / Reasoning
2 DRAG SORT
Solve each inequality, then test if x = 10 is a solution.

Target Categories: x = 10 IS a solution x = 10 is NOT a solution
- x + 2 > 8 (x > 6)
- x − 5 ≥ 3 (x ≥ 8)
- x + 1 ≤ 15 (x ≤ 14)
- x + 3 > 15 (x > 12)
- x − 2 ≥ 9 (x ≥ 11)
- x + 6 < 10 (x < 4)
3 MULTIPLE CHOICE
Final gate — describe the solution set. The vault rule is x > 7. What does the solution set mean?
1. AAll numbers greater than 7, like 8, 9, 10, … (but NOT 7 itself)
2. BOnly the number 7
3. CAll numbers less than 7
4. DEvery number, including 7
✏️ Workspace & Solution Steps

SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]

4 MULTIPLE CHOICE
Clue 1: Which value is a solution of x > 7?
1. A9
2. B7
3. C5
4. D3
✏️ Workspace & Solution Steps
5 MULTIPLE CHOICE
Clue 2: Is 5 a solution of x ≤ 5?
1. AYes, because 5 ≤ 5 is true
2. BNo, because 5 is not less than 5
3. CNo, because ≤ means only smaller numbers
4. DYes, but only because 5 is odd
✏️ Workspace & Solution Steps

SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]

6 ERROR ANALYSIS
Level 2 Extension — Catch the Boundary Mistake
1. 1Vault rule:x > 7
2. 2Test the boundary value:Substitute x = 7: 7 > 7
3. 3Rookie's conclusion:7 > 7 is true, so 7 IS a solution and opens the gate.
4. 4Mark the solution set:The detective adds 7 to the list of solutions.
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation

🗣️ MATHEMATICAL DISCOURSE & TALK MOVES SMP.3 / Construct Viable Arguments

Group Discussion Prompt: How does your visual model justify your mathematical solution?

🗣️ Partner A:
Partner A: "I modeled this by identifying the relationship and..."

👂 Partner B:
Partner B: "I agree with your step because the standard mathematical rule states..."

✍️ The Writing Revolution (TWR) · Sentence Expansion Because · But · So

Complete each sentence stem to demonstrate precise mathematical reasoning:

BECAUSE The ratio remains equivalent because both quantities are scaled by the exact same multiplicative factor.

BUT Two ratios may look similar, but inverting the order of terms fundamentally changes the comparison.

SO The recipe requires 3 parts flour to 2 parts water, so the unit rate is 1.5 cups of flour per cup of water.

📐 CER MATHEMATICAL PROOF MATRIX (CER 2.0) SMP.3 / Proof & Justification

Writing Task: Justify why your mathematical solution is accurate and complete.

C Claim

State your direct mathematical answer/claim with units.

E Evidence

Cite exact numbers, calculations, dimensions, or graph data.

R Reasoning

Explain the mathematical theorem, property, or definition connecting evidence to claim.

✍️ AUTHOR YOUR OWN EXTENSION CHALLENGE

Create an original multi-step word problem aligned to this standard. Include constraints, and write the complete step-by-step mathematical proof below.

✏️ Author Workspace & Complete Solution Key

Student Mastery Self-Assessment:

1 · Need More Support

2 · Getting Closer

3 · Got It / Solid

4 · Master / Can Teach It

Inequality	Inverse Operation	Solution	Graph Description
x + 11 ≥ 20
x − 3 < 7
x + 8 ≤ 20