NapkinCalc

Algebra 1A — Equations & Lines

Inequalities — the one twist

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: inequalities (<, >, ≤, ≥) solve exactly like equations, with one rule to never forget: if you multiply or divide both sides by a negative number, flip the sign. (Check: −2 < 3 is true, but multiply both by −1 and you must write 2 > −3, not 2 < −3.)

Solve −2x + 1 < 9: subtract 1 → −2x < 8; divide by −2 and FLIP → x > −4. The boundary (where the two sides are equal) is x = −4.

bound=9−1−2bound = \frac{9 - 1}{-2} = −4-4 the tipping point: x = −4
✓ pass bound==−4bound == -4 the boundary value
✓ pass −2⋅0+1<9-2 \cdot 0 + 1 < 9 test x = 0 (which is > −4): 1 < 9 ✓ — it is in the solution
✓ pass −2⋅−5+1>9-2 \cdot -5 + 1 > 9 test x = −5 (which is < −4): 11 > 9 — correctly OUT of the solution

Real-world hook: inequalities are budgets and limits — "stay under 2 GB," "earn at least $500," "keep the temperature below 80 °C." The answer is a whole range, not a single value.

Try it yourself: solve −3x > 12. (Remember to flip!) What is the boundary value where −3x = 12?

bndyou=−4bnd_{you} = -4 ✏️ Your turn: find the boundary of −3x > 12 (the x with −3x = 12). The solution is x < −4 — but enter the boundary number here.
✓ pass abs(−3⋅bndyou−12)<10−9\mathrm{abs}\left(-3 \cdot bnd_{you} - 12\right) < 10^{-9} green when −3·(your boundary) = 12