Solve and Graph Inequalities
I can solve an inequality and graph its solution set on a number line.
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🎯 Content Objective / Objetivo de contenido
I can solve an inequality and graph its solution set on a number line.
Today's Flow
Total pacing: ~45 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
Solving an inequality and solving an equation both isolate the variable. What is the same about your steps, and what is different about the answer you get?
Check for Understanding #1
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Case File: Solve and Map It
Detective Kim knows that the suspect withdrew more than $50 from an ATM. The amount plus a $3 fee totaled more than $53. She writes x + 3 > 53 and needs to solve for x, then graph the solution set to share with the bank. How can she find and display all possible withdrawal amounts?
Concept Launch
💡 How do I solve an inequality and then graph it?
I solve an inequality the same way as an equation: use inverse operations to get the variable alone. But the answer is a range of values, so I show all of them on a number line.
Solve with inverse operations, then graph the solution set: open circle for < or >, closed for ≤ or ≥.
Check for Understanding #2
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Now it's your turn
VOCABULARY
⏱ ~8 min
| Term / Término | Meaning / Significado | Example / Ejemplo | Visual |
|---|---|---|---|
| Solve Resolver |
To find the number that makes it true. Encontrar el número que la hace verdadera. |
x + 3 > 10 → subtract 3 → x > 7 — any number greater than 7 works | |
| Graph Graficar |
To show answers on a number line with circles and shading. Mostrar respuestas en una recta numérica con círculos y sombreado. |
x > 7: open circle at 7, shade right — shows all solutions at a glance | |
| Solution Solución |
A number that makes the equation or inequality true. Un número que hace verdadera la ecuación o desigualdad. |
x = 8 is a solution to x > 7 because 8 > 7 is true; x = 5 is not because 5 > 7 is false | |
| Substitute Sustituir |
To put a number in for the letter to check if it works. Poner un número en lugar de la letra para ver si funciona. |
Substitute x = 5 into x + 3 > 10: 5 + 3 = 8, and 8 > 10 is false, so x = 5 is not a solution | |
| Solution set Conjunto solución |
All of the values that make an inequality true. Todos los valores que hacen verdadera una desigualdad. |
For x > 4, the solution set is every number greater than 4, such as 5, 6, 7, ... | |
| Inverse operation Operación inversa |
An operation that undoes another, used to isolate the variable. Una operación que deshace otra, usada para despejar la variable. |
To solve x + 3 > 7, use the inverse of adding 3: subtract 3 to get x > 4. |
Which Word Fits?
To find all the values that make an inequality true is to ___ it.
Use It In a Sentence
Check for Understanding #3
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Turn & Talk — Launch
Solving an inequality and solving an equation both isolate the variable. What is the same about your steps, and what is different about the answer you get?
👂 Listen For
Students note both use inverse operations to isolate the variable, but an inequality's solution is many values (a range) while an equation's is a single value.
Extend: Predict: after you solve an inequality, how will the graph look different from graphing the solution to an equation? Justify.
EXPLORE & PRACTICE
⏱ ~18 min
Visual Modeling Workspace
Use the drawing tray below to annotate the visual model. Teacher: say "Click to reveal" on key steps.
Explore Activity
Solve each inequality using inverse operations, then graph the solution on the number line.
✍️ Explore Discourse
How is solving an inequality similar to and different from solving an equation?
Whiteboard Moment
Show your work clearly. Be ready to explain your thinking to a partner.
Turn & Talk — Explore
What steps did you follow to solve and then graph the inequality, and how did you check a value to make sure your solution set is right?
👂 Listen For
Students isolate the variable, graph the solution set with the correct circle and shading, and substitute a test value to confirm it makes the inequality true.
Extend: If you substitute a value from the shaded region and it makes the inequality false, what does that tell you about your solution? Explain.
Practice Check A
Is x = 5 a solution to x + 4 > 12?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Practice Check B
Solve: x + 6 > 14
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Expression Simplify
Complete the interactive activity using today's strategy.
✍️ Justify Your Thinking
Solve each inequality first, then sort by whether x = 10 is a solution.
A classmate turned in the work below. One step has a mistake. Read every step, find it, name it, and fix it.
Choose ONE option to show what you know — then do it in the workspace below.
Use evidence from today's lesson to complete each frame.
Today's key idea is: "Solve with inverse operations, then graph the solution set: open circle for < or >, closed for ≤ or ≥." — and it works because ___.
Because Solve means ___, but a tricky part is ___, so I have to ___.
A common mistake with Solve is ___. It happens because ___, and the fix is ___.
I can prove my answer is correct by ___, using Graph to check my work.
✍️ TWR · WRITE 3 SENTENCES · 7 MIN
Solve with inverse operations, then graph the solution set: open circle for < or >, closed for ≤ or ≥. because ___
Solve with inverse operations, then graph the solution set: open circle for < or >, closed for ≤ or ≥. but ___
Solve with inverse operations, then graph the solution set: open circle for < or >, closed for ≤ or ≥. so ___
🌱 TWR · GROW THE KERNEL · 6 MIN
Answer these to add detail
Sentence starters (tap to use)
Student Workspace
Solve each inequality using inverse operations, then graph the solution on the number line.
| Column A | Column B |
|---|---|
✏️ Sketch Your Strategy
Differentiation Paths
Step-by-step with a worked model and sentence frames.
Solve: x + 6 > 14
Core practice aligned to the standard.
Extension with error analysis or multi-step reasoning.
Partner Activity
Work with your partner on the practice problems at your differentiation path level. Explain each step using math vocabulary.
Check for Understanding #4
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Real-World Connection
🌍 Math in the Wild
A detective needs at least $75 for a surveillance camera. She has already saved $48. The inequality 48 + s ≥ 75 represents how much more money s she needs.
✍️ Connection Reasoning
How much more does the detective need? How did you solve and graph this inequality?
The detective needs at least $__ more because 48 + s ≥ 75 means s ≥ 75 − ___ = ___.
Turn & Talk — Connect
When would solving an inequality help you stay within a real-life limit?
👂 Listen For
Students give a real budget/time/capacity limit and explain that the inequality's solution set shows all amounts that keep them within the limit.
Extend: Critique: 'The solution to an inequality is just like the solution to an equation — one number.' Use a solved example to explain why this is wrong.
CLOSURE & REFLECT
⏱ ~8 min
Today I learned that ___ because ___.
One thing I am still not sure about is ___.
Solve and describe the graph: x + 5 < 11
Bonus Exit Check
Solve: x − 9 ≤ 3
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Reflection & Self-Assessment
Continue Learning
Launch the Full Interactive Activity
Students continue practice in the HTML lesson engine with auto-check, hints, and differentiation.
Family Connection
Share tonight's family homework and discuss one vocabulary word at home.
Open Family Homework ↗Teacher Notes
⏱️ Pacing Guide
- Launch & vocab: 12 min
- I Do / We Do / You Do: 15 min
- Explore & practice: 15 min
- Connect & closure: 8 min
Total: ~45 min
🎯 Listen For · Common Errors
• Students note both use inverse operations to isolate the variable, but an inequality's solution is many values (a range) while an equation's is a single value.
• Students isolate the variable, graph the solution set with the correct circle and shading, and substitute a test value to confirm it makes the inequality true.
• Students give a real budget/time/capacity limit and explain that the inequality's solution set shows all amounts that keep them within the limit.
• Listen for students naming a specific strategy tied to 6.EE.5 — not just "I multiplied." They should connect steps to the key idea.
Common mistake: A common mistake in Solve and Graph Inequalities is skipping the key idea: "Solve with inverse operations, then graph the solution set: open circle for < or >, closed for ≤ or ≥." — always check your work against this rule before you submit.
Answer Key (Teacher Appendix)
Hide this slide during presentation or move to the end of your copy.
✓ Practice 1: No, because 5 + 4 = 9, and 9 is not greater than 12 — Substitute: 5 + 4 = 9. Since 9 > 12 is false, x = 5 is NOT a solution.
✓ Practice 2: x > 8 — Subtract 6 from both sides: x > 14 − 6 = 8. Graph: open circle at 8, shade right.
✓ Practice 3: x ≤ 12 — Add 9 to both sides: x ≤ 3 + 9 = 12. Graph: closed circle at 12, shade left.
✓ Practice 4: x ≥ 15 — Subtract 10 from both sides: x ≥ 25 − 10 = 15. Graph: closed circle at 15, shade right.
✓ Exit ticket: x < 6; open circle at 6, shade left — Subtract 5 from both sides: x < 11 − 5 = 6. Open circle at 6 (not included), shade left. ✓