9. Break-Even and Comparison Problems

A break-even point is where two costs, earnings, or quantities are equal.

Example: Plan A costs \(20+5m\). Plan B costs \(50+2m\), where \(m\) is months. To find when they cost the same:
\[ 20+5m=50+2m. \]
Subtract \(2m\):
\[ 20+3m=50. \]
Subtract 20:
\[ 3m=30,\qquad m=10. \]
At 10 months, the plans cost the same.

Before 10 months, one plan may be cheaper. After 10 months, the plan with the smaller rate may become cheaper.

Common misconception: choosing the plan with the lower starting cost without considering the monthly rate.

Academic habit: for comparison problems, write one expression for each option, then set them equal only when the question asks when they match.