a To construct a quadratic inequality whose solutions are all real numbers, we need to set a statement which will always be true. Recall that the graph of the parent quadratic function y=x^2 is a parabola.
We see above that the parabola opens upwards and that the vertex is on the x-axis. Therefore, the y-values are all non-negative. We can translate the parent function vertically upwards by adding a positive constant. For example, f(x) = x^2+2 will represent the same graph as the parent function but translated 2 units up.
Since all y-values are now positive, the inequality x^2+2 > 0 will hold true for all x-values, as required. Notice that this is just an example solution, as there are infinitely many inequalities satisfying this condition.
b To find a quadratic inequality whose solution set is the empty set, we need to set a statement which will always be false. Recall that the parent quadratic function y=x^2 does not take negative values, as any negative number becomes positive after being squared.
The inequality x^2<0 will not be true for any x value, as required. Notice that this is just an example solution, as there are infinitely many inequalities satisfying this condition.
