Part 1 — Write, Then Solve

The council's goal is $200. So far they have raised $45. President Aria translated the words into an equation: 45 + x = 200, where x is the amount still needed.

Subtract 45 from both sides: x = 200 − 45 = $155 still needed. Check: 45 + 155 = 200. ✓

Analyze — Q1

Why is writing the equation before solving helpful here?

Solve It — #1

They raised $60 toward a $250 goal. Write 60 + x = 250 and solve for x.

x = $

Part 2 — How Many to Sell?

Tickets sell for $8 each. The council needs $96 from ticket sales. Equation: 8x = 96, where x is the number of tickets.

Divide both sides by 8: x = 96 ÷ 8 = 12 tickets. Check: 8 × 12 = 96. ✓

Solve It — #2

Bracelets sell for $5 each, and they need $75. Solve 5x = 75 for the number of bracelets.

x =

Part 3 — An Inequality for the Goal

"We don't just want exactly $200 — we want at least $200," Aria said. That is an inequality: x ≥ 200. Any amount of 200 or more is a success.

x ≥ 200 means x can be 200, 201, 250, … The smallest whole-dollar amount that meets the goal is $200. Values like $180 do not satisfy it.

Analyze — Q2

Why use x ≥ 200 instead of x = 200?

Solve It — #3

A prize needs at least 30 cans (c ≥ 30). What is the smallest whole number of cans that wins the prize?

Smallest c:

Part 4 — Solve and Verify

For the snack stand, the council wrote 5x = 60 (x = number of combo boxes at $5). Aria solved it and verified by substitution — a habit good problem-solvers keep.

x = 60 ÷ 5 = 12. Verify: substitute back, 5 × 12 = 60. ✓ True, so 12 is correct.

Solve It — #4

Solve 7x = 84, then verify by substitution. What is x?

x =

Analyze — Q3

Why does Aria substitute her answer back into the equation?

After You Read — Analytical Writing

Make a Mathematical Argument

Prompt A — Model a goal. Write your own one-step equation for a real fundraiser situation, solve it, and verify by substitution. Explain why each step keeps the equation balanced.

Prompt B — Equation vs. inequality. Explain the difference between "raise exactly $200" (x = 200) and "raise at least $200" (x ≥ 200). Give an amount that satisfies one but not the other, and say why.

Optional academic frame: "The equation models ______; solving gives ______, which I verified by ______."

Write your argument here:

Challenge Extension

Think Further

The council needs at least $200 and sells tickets at $8 each. Write an inequality for the number of tickets t, and find the fewest whole tickets they must sell to reach the goal. Explain your reasoning. (Try it, then check with your teacher.)

How You Are Scored

Rubric

Category	4 — Advanced	3 — Proficient	2 — Developing
Comprehension & inference	All analysis questions correct; explains equation vs. inequality & verifying	2 of 3 correct	1 correct
Multi-step math	All 4 Solve-It answers correct (write+solve, solve, inequality, solve+verify)	3 correct	2 correct
Mathematical argument	Models a situation; solves, verifies, and explains balance/inequality	Claim with some evidence	Little evidence