Higher-degree inequalities

Order the four roots and check the product sign between and outside them. A non-strict inequality includes zero.

Factor out x and factor the difference of squares. Find the valid part of the interval, then count its integers, excluding any root.

Solve the cubic inequality using signs on intervals. Intersect its solution set with the second condition and retain valid endpoints.

Both squares are non-negative. Determine when their product can be at most zero by checking when either factor is zero.

Let t = x². Solve the quadratic inequality in t, then recover x, retaining only x > 0.

An even power removes a negative factor sign; an odd power preserves it. Check factor signs in each interval and count negative products.

Relate the fourth power to distance |x − a|. Take the positive fourth root of the right side and express the distance bound as an interval.

Mark the three roots and determine factor and product signs in the four intervals. A strict inequality excludes the roots.

The squared factor is positive except at its root. The other factor determines the sign, but the strict inequality excludes the double root too.

Record the original denominator restriction before cancellation. Solve the remaining inequality and count valid integers, excluding the forbidden value.