Solve for the Unknown: Equations & Inequalities
You just got hired as the logistics planner for the Maker Market — the school's pop-up shop. Boxes, budgets, and table space all hide an unknown number. Your job is to write an equation or inequality, then solve it to keep the market running. Eres el planificador del mercado escolar. Tu trabajo es escribir y resolver ecuaciones y desigualdades.
1 Engage
Hook & essential question
A shipment arrives for the Maker Market. The pan balance below holds one mystery box (we call its weight x) plus a 3-pound weight on the left. The right pan holds a 10-pound weight. The pans are level — perfectly balanced.
The balance is level, so the two sides weigh the same.
Quick think (no calculator): If the box plus 3 pounds equals 10 pounds, how heavy is the mystery box? Write your guess and one sentence explaining how you got it. You will check it in the Apply section.
Essential question: How can I turn a real situation into an equation or inequality, and then undo the steps to find the unknown?
2 Explore
Investigate before the lesson
Explore each tool below for a few minutes. As you go, watch for one big idea: to keep an equation true, whatever you do to one side you must do to the other side too.
Try it: keep the balance level
The left pan is x + 3. The right pan is 10. Slide to change x. The pans stay level only when both sides are equal. Mantén la balanza nivelada.
3 Explain
The math, in plain language
An equation says two amounts are equal (it has an = sign). A variable like x stands for the unknown number. To solve a one-step equation, you undo the operation using its inverse — keeping both sides equal, just like a level balance.
- variable
- a letter that stands for an unknown number · la variable
- equation
- two expressions joined by = (they are equal) · la ecuación
- inverse operation
- the opposite move (+ undoes −, × undoes ÷) · la operación inversa
- inequality
- a comparison using <, >, ≤, or ≥ · la desigualdad
- solution
- the value that makes it true · la solución
Addition equation
x + 3 = 10 → x = 7
Subtract 3 from both sides. 10 − 3 = 7. Adding was the problem, so subtracting is the inverse.
Subtraction equation
x − 4 = 9 → x = 13
Add 4 to both sides. 9 + 4 = 13.
Multiplication equation
5x = 35 → x = 7
Divide both sides by 5. 35 ÷ 5 = 7.
Division equation
x ÷ 6 = 4 → x = 24
Multiply both sides by 6. 4 × 6 = 24.
Worked example — the mystery box
Back to the Engage balance: x + 3 = 10.
Subtract 3 from both sides: x + 3 − 3 = 10 − 3
So x = 7. The box weighs 7 pounds.
Check: put 7 back in → 7 + 3 = 10 ✓. (How close was your Engage guess?)
Inequalities tell you a range, not one answer
Each market table fits at most 8 boxes. If n is the number of boxes, then n ≤ 8. That is true for 0, 1, 2, … all the way up to 8. We show every solution on a number line.
A closed (filled) dot means "and including" (≤ or ≥). An open (hollow) dot means "but not" (< or >).
Level 1 · support Step-by-step helper / Ayuda paso a paso
- Find the variable (the letter) and what is happening to it (+, −, ×, or ÷).
- Do the inverse operation to both sides. + undoes −. × undoes ÷.
- The variable should now be alone. Write x = your number.
- Check: put your number back in. Does the equation stay true? Comprueba tu respuesta.
Level 2 · enrichment Push your thinking
The inequality n ≤ 8 has many solutions, but in the real market n must also be a whole number that is 0 or more. How many real values of n actually work? Explain why a number line and the real situation can disagree about which solutions count.
4 Apply
Show what you can do — auto-checked
Level 1 · support Sentence starters for explaining
"The variable is ____. The operation is ____, so the inverse is ____. I did ____ to both sides. My answer is x = ____, and when I check it the equation is still true."
"La variable es ____. La operación es ____, así que la inversa es ____. Hice ____ en ambos lados. Mi respuesta es x = ____."
Level 2 · enrichment Write the story
Write your own Maker Market word problem that solves to x = 9 using a multiplication equation, AND a separate situation that graphs to x < 5. Trade with a partner and solve each other's.
Teacher Notes & Answer Key (not printed)
Stage-by-Stage Notes (Engage → Explore → Explain → Apply → Reflect)
- Engage: a balance puzzle motivates "do the same to both sides."
- Explore: solve one-step equations with inverse operations.
- Explain: formalize inverse operations and inequality notation.
- Apply: answer key below.
- Reflect: students explain checking a solution by substitution.
Apply — Answer Key
- Q1 — x + 3 = 10: x = 10 − 3 = 7.
- Q2 — equation result: 6.
- Q3 (MC): answer c.
- Q4 — "at most 8" inequality (MC): answer b (≤ 8).
- Q5 — "more than 12" (MC): answer d (> 12).
- Q6 — divide evenly: 3.
Standard
CCSS 6.EE.B.5–8.
5 Reflect
Think about your thinking
A. Why must you do the same operation to both sides of an equation? Use the balance idea in your answer.
B. An equation usually has one answer, but an inequality has many. Explain the difference in your own words.
C. One thing that is now clear to me / Una cosa que ahora entiendo bien:
Your reflections save with this HyperDoc when you use Save as PDF or Save as DOC above.
6 Extend
Optional challenge project
Plan the Maker Market budget
You have a budget and some unknowns to solve. On paper or a slide, do all four:
- The market spent $48 on supplies, split equally across 6 booths. Write and solve an equation for the cost per booth.
- Each booth needs at least 5 posters. Write an inequality for the number of posters p, and graph it on a number line.
- Tickets sell for $4 each. The market wants more than $200. Write an inequality for the number of tickets t.
- Check every answer by substituting it back in.
Planifica el presupuesto del mercado: escribe y resuelve ecuaciones, y grafica desigualdades en una recta numérica.