A Budget That Shrinks

You have 50 and you spend 4 a day. For how many days will you still have at least 10? Call the number of days $x$. After $x$ days you have $50-4x$. "At least 10" means 10 or more, so the sign is ≥:

$$50-4x \ge 10$$

$$-4x \ge -40$$

$$x \le 10$$

Dividing by -4 flips the sign. The answer is that you can go on for up to 10 days. Check the answer against the story. More days means less money, so the condition sets an upper limit on the days, not a lower limit.

A quantity that goes down has a negative coefficient on $x$. When you divide by that negative coefficient to get $x$ alone, reverse the inequality sign. Then check that the range you found fits the story.