Solving linear equations one legal move at a time, and naming the property of equality behind each move; the distributive property, like terms and variables on both sides; equations with one solution, no solution or every number as a solution; inequalities and the one rule that flips the symbol; compound inequalities on a number line. Then the modeling side: writing an equation or inequality from a phone plan, a gym fee or a bus trip, saying in a sentence what the solution means, and rearranging a formula for any of its letters. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
Whatever x was, it got multiplied by 2, then 3 was added. That is the order it was built in.
Undo in the reverse order of operations. It was built multiply-then-add, so you take it apart subtract-then-divide. Every legal move you make is doing the same thing to both sides.
What people get wrong
⚠️People often think…
3(x − 5) is 3x − 5.
Two terms inside means two products out: 3x − 15. The 3 does not get tired after the first one.
Count the terms inside. That is how many products you owe.
⚠️People often think…
To solve 2x + 3 = 11, divide 11 by 2 first.
The +3 happened last, so it comes off first. Subtract 3 to get 2x = 8, then divide. Dividing first would mean splitting the 3 as well, and it is not attached to the x.
Last on, first off.
⚠️People often think…
−2x > 8 gives x > −4.
Dividing by a negative mirrors the number line, so the symbol turns around: x < −4. Check it with a number — x = −5 works, x = 0 does not.
Test one number. It costs five seconds and catches this every time.
⚠️People often think…
“7 less than 2n” is written 7 − 2n.
“Less than” reverses the order: it is 2n − 7. You are taking 7 away FROM 2n. The same flip happens with “subtracted from” and “fewer than.”
Less THAN and subtracted FROM both flip. Rewrite before you solve.
Undo, in reverse
1Watch one
Solve 2x + 3 = 11.
The +3 went on last, so take it off first.
2x = 8.
Now undo the × 2.
x = 4. Check: 2(4) + 3 = 11.
2Do one with me
Solve 3(x − 5) = 21.
Distribute: 3x −
Add 15 to both sides: 3x =
Divide by 3: x =
💬One sentence, then you move on
Why do you undo an equation in the reverse order it was built?
3Try one
Write “7 less than 2n” as an expression.
I want a hint first
Say it out loud as an action: take 7 away FROM 2n. The thing you take from is written first.
💬Last one — then you're done here
What happens when the variable cancels and you are left with 6 = 6?
Where this goes
Where this lives
Any time you know the end result and need the starting number: a total bill working back to the hourly rate, a final grade working back to what you need on the test.
What this feeds
Next unit is functions and linear models, where the same equation gets a name and a graph.
Write one real situation where you know the answer and need the starting amount.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🪜 Solving a linear equation — the routine
Distribute and tidy each side (like terms) → gather the variable terms on one side and the constants on the other, moving the SMALLER variable term → undo the add-or-subtract, then the multiply-or-divide → substitute into the ORIGINAL equation and evaluate each side separately. Write the property beside every step: distributive, addition/subtraction property of equality, division property of equality.
Tidy, gather, undo, check.
💡 The whole unit in one idea
An equation is a balance, and every legal move keeps it balanced: whatever you do to one side you do to the other, so the new equation has exactly the same solutions as the old one. Inequalities obey the same rules with one exception — multiplying or dividing both sides by a negative number reverses the symbol. Everything else (formulas, word problems, compound inequalities) is the same balance wearing a different outfit.
⚠️ Traps the test loves
Distributing to only the first term: 3(x − 5) written as 3x − 5. Two terms inside, two products out.
Dividing before subtracting: solving 2x + 3 = 11 by dividing 11 by 2 first. Undo in reverse order of operations.
Forgetting to flip when dividing an inequality by a negative: −2x > 8 is x < −4, not x > −4.
Answering “x = 0” when the variable cancels. Read what is left: 6 = 6 means every number; 1 = 5 means no solution.
Translating “7 less than 2n” as 7 − 2n. It is 2n − 7 — “less than” reverses the order.
Rearranging P = 2l + 2w to w = P − 2l and stopping. The whole side still has to be divided by 2.
📋 Words, symbols and solution sets — the chart
Words
Symbol
Boundary included?
Number line
more than · over · exceeds
>
no
open circle, shade right
less than · under · fewer than
<
no
open circle, shade left
at least · no less than · minimum
≥
yes
closed circle, shade right
at most · no more than · maximum
≤
yes
closed circle, shade left
x’s cancel, TRUE left over
identity
—
all real numbers
x’s cancel, FALSE left over
contradiction
—
no solution (∅)
🎯 How the test will ask
A multi-step equation with parentheses and variables on both sides — “solve and check.” Show the check as two separate calculations.
A worked solution with blanks — “which property justifies this step?” Name the operation you see between the two lines.
Three equations — “one solution, none, or infinitely many?” Compare the x-coefficients first, then the constants.
An inequality with a negative coefficient — “solve and graph.” Watch for the flip; check a number from your shaded side.
A phone plan, gym or taxi story — “write an inequality, solve it, and say what the solution means.” The answer is a sentence with units.
A formula — “solve for the indicated variable.” Treat the other letters as numbers; parentheses around the whole side before dividing.
A student’s wrong work — “find the error.” Test their answer in the original equation first; then look for the bad step.
✅ Before the test, can you…
Solve an equation with parentheses, fractions or decimals, and variables on both sides, and name the property for each step?
Check a solution by evaluating both sides of the ORIGINAL equation?
Tell an identity from a contradiction without solving all the way?
Solve and graph an inequality, flipping only when you multiply or divide by a negative?
Solve a compound inequality by doing the same thing to all three parts?
Write an equation or inequality from a story, solve it, and explain the solution in context with units?
Rearrange a formula such as P = 2l + 2w or F = 1.8C + 32 for any one of its letters?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: put a solution in order and name the property behind every step, decide whether an equation has one solution, none, or all of them, argue what a number means for a phone bill, then rearrange a formula one legal move at a time. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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Lab
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Explained in own words
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The answer key is for a teacher: it prints only from here, for the unit on screen. Print the study packet prints the study pages and a blank quiz — never the answers.