how is the solution to an inequality different from that of an equation
For functional inequality (two variables) they are very similar, with the equality determining the border of the inequality. Graphed, the equality forms the edge of the shaded region that is the inequality.
Algebraically, the only thing you have to worry about for the inequality is the direction of the sign. It changes if you multiply or divide both sides of the equation by a negative number. Otherwise, it is solved very similar to an equality.
However, if the inequality is just a single variable, then things get very different. The equality determines points on the number line that form the endpoints of the inequality. But, you must do some test to determine which intervals are included in the inequality. For example: