I’m getting tripped up by solving linear equations when there are parentheses and minus signs floating around. It reminds me of trying to balance grocery bags on both arms-every time I shift one can of beans, something else tips over.
Here’s the one I’m wrestling with right now:
3(x – 4) + 2 = 2x – (x – 1)
My attempt (which I think is right, but I’m totally second-guessing myself):
– Distribute and deal with the minus sign: 3x – 12 + 2 = 2x – x + 1. I flipped both signs inside the second parentheses because of the minus outside-does that part make sense?
– Combine like terms: 3x – 10 = x + 1
– Subtract x from both sides: 2x – 10 = 1
– Add 10 to both sides: 2x = 11
At this point I feel like I know what to do next, but this is exactly where Past Me (high school flashbacks!) would mess up a sign or do that sketchy “move across and change the sign” thing without thinking. I’m trying to stick to the “do the same thing to both sides” idea like a balance scale, but I still hesitate.
Two things I want help with:
1) Are each of those steps actually legit, especially the part where I handled the minus in front of (x – 1)?
2) Is there a simple way to remember when signs flip and when they don’t-like a plain-English or real-world analogy that sticks? I keep mixing up whether I’m subtracting from both sides or just moving a term and magically changing its sign.
I’ve struggled with this before on homework and ended up with totally bonkers answers because I lost a minus somewhere. Would love a sanity check on the steps above and a sticky mental rule so I stop derailing at the same spot!
Ready to make maths more enjoyable, accessible, and fun? Join a friendly community where you can explore puzzles, ask questions, track your progress, and learn at your own pace.
By becoming a member, you unlock:
3 Responses
You handled everything just right. Distributing: 3(x − 4) = 3x − 12, and the tricky bit −(x − 1) really does flip both signs because it’s like multiplying the whole parenthesis by −1, so −(x − 1) = −x + 1. That gives 3x − 12 + 2 = 2x − x + 1, which simplifies to 3x − 10 = x + 1. Subtract x from both sides (think: take x off each “arm” of the balance) to get 2x − 10 = 1, then add 10 to both sides to get 2x = 11, and finally divide by 2: x = 11/2. If I didn’t miscopy anything, that’s spot on-and a quick check works: LHS = 3(11/2 − 4) + 2 = 13/2 and RHS = 2(11/2) − (11/2 − 1) = 13/2.
A sticky way to remember the sign business: a minus sign in front of parentheses is like putting a “reverse” sticker on the whole grocery bag-you reverse the sign of every item in that bag. And the “move across and change the sign” mantra is really just shorthand for “add the opposite to both sides.” For example, to “move” x from the right to the left, you subtract x from both sides; to “move” −10 from the left to the right, you add 10 to both sides. Nothing magical is happening-just keeping the scale balanced with equal-and-opposite actions. If you want a quick refresher with more examples, this Khan Academy lesson on solving linear equations walks through exactly these steps: https://www.khanacademy.org/math/algebra/one-variable-linear-equations/alg1-solve-equations/v/solving-linear-equations
You nailed it: distributing gives 3x−12+2 = 2x−x+1 → 3x−10 = x+1, then subtract x and add 10 to get 2x=11 so x=11/2; the “−(x−1)” flips both signs because you’re multiplying that whole parenthesis by −1. Sticky rule: don’t “move” terms-add/subtract the same thing to both sides (balance scale vibes), and read −(stuff) as “add the opposite,” so everything inside changes together-want to try a similar one like 4(2−y)−3 = y−(3−y)?
You handled the parentheses correctly. Distribute first: 3(x − 4) + 2 = 3x − 12 + 2 and 2x − (x − 1) = 2x − x + 1, because the minus out front is like multiplying by −1. That gives 3x − 10 = x + 1. Subtract x from both sides to keep it balanced: 2x − 10 = 1. Add 10 to both sides: 2x = 11, so x = 11/2. I sometimes rewrite −(x − 1) as −x − (−1) = −x + 1 to remind myself why the constant ends up positive; strictly speaking, a minus outside “flips” the signs inside. (Some people say only the first term flips, but that actually misses the constant-so I’d stick with distributing −1 to each term.)
A memory trick: instead of “move it across and change the sign,” say “add the opposite to both sides.” That’s what “moving” is really doing, and it prevents random sign flips. Signs change only because of two legal moves: (1) you added/subtracted the opposite on both sides, or (2) you distributed a −1. If you multiply both sides by −1, everything on that side changes sign together; you wouldn’t selectively flip signs inside a parenthesis unless you’re distributing that −1 through it. For a longer walkthrough, see Khan Academy’s overview of solving equations with the distributive property: https://www.khanacademy.org/math/algebra/one-variable-linear-equations/alg1-distributive-property-equations/v/solving-equations-using-distributive-property
Quick example: Solve 4 − (2x − 3) = x + 5. Distribute the minus: 4 − 2x + 3 = x + 5, so 7 − 2x = x + 5. Subtract x from both sides: 7 − 3x = 5. Subtract 7: −3x = −2, so x = 2/3. Here I “moved” the −2x by adding 2x to both sides (that’s the same as flipping its sign when it crosses), and the earlier sign flip came only from distributing the − through the parentheses.