For an equation, there is no direction, both sides of the equation are always equal. For an inequality, however, one side will always be greater or less than the other side. When you multiply or divide an inequality by a negative number, you need to change the direction of the sign. For example, if the inequality is -4x > 8, and you divide the equation bye -4 to solve it, it would be x < -2.
In an equation, the answer you arrive to will be an discrete value. For example, in the equation 4x = 8, the answer is x = 2. The answer is simply 2. In an inequality, the answer is not a discrete value, but a range. For example, in the inequality 4x > 8, the answer is x > 2. The answer is not 2, but rather any number greater than 2.
In an equation, to prove the answer is true, all you need to do is plug the answer into the equation. For example, if the equation is 4x = 8 and the answer is x = 2, all you need to do is prove the equation is plug in 2: 4(2) = 8. The answer is true. However, in some inequalities, you will have a different range of answers, so to prove the answer you need to plug in more numbers. For example, if the equation is x^2 > 9, and the answer is x<-3 and x>3, which means the answer is less than -3, or greater than 3. To prove this you need to do more steps. You would check 4, for a number greater than 3: 4^2>9 is 16>9, the answer is true. For -4, (-4)^2 > 9 is 16 > 9, which is true. For 1, 1^2 > 9 is 1 > 9, the answer is not true. Therefore, x < -3 and x > 3 is the correct answer.
In an equation, you will only have one answer, x = b. In an inequality, especially with absolute values, you can have multiple answers. For example, for the inequality |x| > 2, then the answer would be x > 2 or x < -2. Rather than the one answer, you have two that could be true, likewise, it could be a discrete range, such as a < x < b.