Q.Represent the following inequalities in the interval notation:

A linear inequality is solved exactly like a linear equation, with ONE crucial exception: multiplying or dividing both sides by a negative number reverses the inequality's direction; multiplying by a positive number never does.

Systems ('and'/'or'). Combining two inequalities with 'and' takes the intersection of their solution sets; combining with 'or' takes the union -- and when one solution ray is already a subset of the other, the union collapses to just the larger one (e.g. '$x<-1$ or $x<3$' is simply $x<3$).

Clearing fractions. Multiply both sides by the LCM of all denominators (a positive number, so no flip) before isolating the variable.

Word problems translate directly into these inequalities: an 'average $\ge$ target' condition becomes 'total $\ge$ target $\times$ count'; a 'more than / less than' comparison between two pay or pricing schemes becomes a direct inequality between the two formulas; a percentage-range requirement (like an acid mixture staying between two concentration bounds) becomes TWO separate inequalities, solved and then intersected.

Restricting to a number system. Once the real-number solution interval is found, restricting to natural numbers or integers means keeping only the points of that interval that belong to $N$ or $Z$ -- take care with which direction to round at a non-integer boundary.