Algebra · Inequalities
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Linear inequalities
Why inequalities solve like equations except for one rule — flipping on a negative multiplier — why the answer is a RANGE not a number, and how double inequalities and integer lists work.
Algebra · Inequalities
Linear inequalities
Why inequalities solve like equations except for one rule — flipping on a negative multiplier — why the answer is a RANGE not a number, and how double inequalities and integer lists work.
Why it works
An inequality is a statement about a range: asks which values of keep the left side below 17. Solve it with the same balance moves as an equation — add, subtract, multiply, divide both sides — and the answer comes out as a range too:The answer IS "" — every number below 5. Writing "" answers a different question; the inequality symbol must survive to the final line.
The one new rule: multiplying or dividing by a NEGATIVE flips the symbol. Watch why: is true, but multiply both sides by and — the order reversed, because negating reflects the number line. So for : subtract 7 to get , then dividing by flips: . (Dodge available: add to both sides instead — — and no flip is ever needed.) Adding and subtracting NEVER flip.
Translate the words precisely. "At least 6" means ; "at most" means ; "more than" is strict . One symbol is one mark.
Double inequalities are two constraints handled at once. has THREE parts — whatever you do, do to all three:
Integer lists read the symbols carefully. The integers with are : the strict EXCLUDES 1, the inclusive KEEPS 4. For : . Check both ends against their own symbol — that's where the marks leak.
Inequalities model budgets and limits. "Van hire costs £40 plus £22 per day; Amy can spend at most £150": , so and — at most 5 days. The formed inequality is marked exactly like a formed equation.