In quadratic inequalities, if the inequality is greater/less than equal to 0, what must be included in the final answer?

In quadratic inequalities, when the inequality is expressed as greater than or less than or equal to zero, it signifies the need to identify the values for which the quadratic expression evaluates to zero or produces negative values.

When dealing with the inequality that states the expression is less than or equal to zero, the solution involves determining the range of x-values that either yield a value of zero (the roots of the quadratic) or are such that the quadratic evaluates to negative numbers. Thus, it's important to include points where the expression equals zero as part of the solution set.

For inequalities that are greater than or equal to zero, it similarly applies: the values that make the quadratic expression positive or equal to zero must be included. This means that in either case, acknowledging the values where the quadratic equals zero is essential to accurately represent the solution to the inequality.

In summary, the condition of the inequality dictates that zero itself, along with any values that lead to a negative response for “less than” cases or a positive response for “greater than” cases, must be incorporated into the final answer. This clearly illustrates why including the threshold (greater/less than equal to zero) is critical to capture the complete solution set for the quadratic inequality.