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3 Responses
Love this kind of puzzle-feels like balancing a see-saw at the park! Your distribution is spot on: 3(2x − 5) − 4 becomes 6x − 15 − 4 = 6x − 19, so no sign got lost there (I picture “owe 15 more 4” = owe 19). Then moving terms: subtract 2x to get 4x − 19 = 8, add 19 to get 4x = 27, so x = 27/4 = 6.75. A super-quick check is to plug it back: LHS = 3(2·6.75 − 5) − 4 = 21.5 and RHS = 2·6.75 + 8 = 21.5-nice match. I also like a sanity check: try x = 7-LHS is 23 and RHS is 22, so the true x should be a bit less than 7, which fits 6.75. Sometimes I even think of “folding” the −4 into the parentheses like 3(2x − 9) to keep track of constants, and then the 2x + 8 on the right sort of “catches up,” which kind of explains why the −19 shows up-but that mental shortcut is a bit hand-wavy. Bottom line: your constants are fine, and your steps lead to the right answer. Hope this helps!
You’re fine-no sign lost. Distribute first: 3·2x = 6x and 3·(−5) = −15, then subtract 4 to get 6x − 19, so 6x − 19 = 2x + 8. Move terms: subtract 2x and add 19 to get 4x = 27, so x = 27/4. Quick check: plug it in-LHS becomes 3(27/2 − 5) − 4 = 51/2 − 4 = 43/2, RHS is 27/2 + 8 = 43/2, so it matches. Faster mental trick: for ax + b = cx + d, x = (d − b)/(a − c). Here that’s (8 − (−19)) / (6 − 2) = 27/4. When I was first grinding through these, I kept dropping a minus after distributing; what fixed it was lumping the constants right away: 3(2x − 5) − 4 = 6x − (15 + 4) = 6x − 19, so there’s no rogue sign to forget.
You’re actually spot on! Distribute first: 3(2x − 5) gives 6x − 15, and then the “− 4” makes it 6x − 19 on the left, so 6x − 19 = 2x + 8. Move the x’s: subtract 2x to get 4x − 19 = 8, then add 19 to get 4x = 27, so x = 27/4 (which is 6.75). A quick way to check is to plug it back: LHS = 3(2·27/4 − 5) − 4 = 3(27/2 − 5) − 4 = 3(17/2) − 4 = 51/2 − 4 = 43/2, and RHS = 2·27/4 + 8 = 27/2 + 16/2 = 43/2, so it matches-no sign dropped. A small tidying trick to avoid the “−19” is to add 4 to both sides first: 3(2x − 5) = 2x + 12, then distribute to get 6x − 15 = 2x + 12, and proceed to 4x = 27 just as before. As a mini example with the same idea: solve 2(3x − 4) − 5 = x + 7. Add 5 first to get 2(3x − 4) = x + 12, distribute to 6x − 8 = x + 12, then 5x = 20, so x = 4; a quick plug-in confirms it. You’ve got good instincts-your constants were handled correctly.