Inequalities using multiplication and division

Multiplying or dividing an inequality by a positive number is safe — the direction stays the same.
But by a negative number, the inequality sign must be reversed.
This is the single most important rule for solving inequalities.

Here is why the flip happens.
Start with $3 > 2$ and multiply both sides by $-1$: you get $-3$ and $-2$.
But $-3 < -2$, so the sign had to flip to stay true.

Dividing by a positive number needs no change.
From $4J \ge 49$, divide both sides by $4$: $J \ge \frac{49}{4} = 12.25$.
The sign stays $\ge$.

Dividing by a negative number needs the flip.
From $-7x > 1$, divide both sides by $-7$ and reverse the sign: $x < -\frac{1}{7}$.
Forgetting to flip is the most common inequality mistake, so reverse the sign every time you divide or multiply by a negative.

Solve $-3x > 12$.
Divide both sides by $-3$, which is negative, so flip the sign: $x < \frac{12}{-3}$.
So $x < -4$.

Solve $5y \ge 35$.
Divide both sides by $5$, a positive number, so the sign stays: $y \ge \frac{35}{5}$.
So $y \ge 7$.

Solve $-\frac{x}{2} > 4$.
Multiply both sides by $-2$, which is negative, so flip the sign: $x < 4 \times (-2)$.
So $x < -8$.