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3 Responses
You’re not overthinking it-you only flip the inequality when you multiply or divide by a negative, not when you add or subtract. So 3 − 2x ≥ 7 becomes −2x ≥ 4 after subtracting 3 (no flip yet). Now you divide by −2, and that’s the annoying part: you must flip ≥ to ≤, so x ≤ −2. Quick sanity check with a tiny example: 2 < 5 is true, but multiply both sides by −1 and you get −2 > −5-the sign flips because negatives reverse the order on the number line. If you hate remembering that, here’s a trick: shuffle things to keep the x-coefficient positive before dividing. From 3 − 2x ≥ 7, add 2x to both sides and subtract 7 to get −4 ≥ 2x, then divide by 2 (no flip) to get −2 ≥ x, which is the same as x ≤ −2. Same answer, fewer booby traps.
Yes-subtracting 3 does not change the inequality, but dividing by −2 does. From 3 − 2x ≥ 7, subtract 3 to get −2x ≥ 4. Now divide by −2 and flip the sign, giving x ≤ −2. A quick check: x = −3 works (3 − 2(−3) = 9 ≥ 7), and x = 0 doesn’t (3 ≥ 7 is false), so the direction is right.
Simple example: −4t ≤ 12 implies t ≥ −3 for the same reason-the division is by a negative, so the sign reverses. As a mental shortcut, I sometimes think of “moving” −2x to the other side as flipping the arrow once (3 ≥ 7 + 2x), then dividing by +2 keeps it (−2 ≥ x), which matches x ≤ −2. That shortcut is a bit mixed up about why the flip happens, but it lands on the same answer.
You’re not overthinking it-this is exactly the spot where the sign flip matters. Adding or subtracting the same number on both sides keeps the inequality pointing the same way, but multiplying or dividing by a negative flips it, because it reverses the order on the number line (for example, 2 < 5 but −2 > −5). For your problem: 3 − 2x ≥ 7 → subtract 3 to get −2x ≥ 4 → now divide by −2 (a negative), so flip the sign: x ≤ −2. A quick check: x = −3 gives 3 − 2(−3) = 9 ≥ 7 (true), while x = −1 gives 5 ≥ 7 (false), so x ≤ −2 is right. If the flipping step makes you nervous, you can dodge it by keeping divisions positive: 3 − 2x ≥ 7 → add 2x to both sides: 3 ≥ 7 + 2x → subtract 7: −4 ≥ 2x → divide by +2: −2 ≥ x, which is the same as x ≤ −2. Simple example to remember the rule: starting from 1 < 3, dividing both sides by −1 gives −1 > −3-the inequality flips because of the negative.