Quadratic Inequalities

Quadratic Inequalities

GCSE(H),

Inequalities may also be represented in quadratic form. To represent an inequality in quadratic form, plot the graph for the equation equivalent of the quadratic, then determine which side of the line the inequality applies to. Given the nature of a quadratic, this will be either inside or outside the loop.

As with all inequalities, the inequality itself may be less than, less than or equal to, greater than, or greater than or equal to.

A quadratic inequality and a linear inequality are often linked together, giving a bounded area.

Examples

1. The graph below shows an inequality `y > -x^2 + 5`. Indicate which, if either, of the points A and B satisfy that inequality.

Answer: Point B

Substitute the coordinates for point A into the inequality: `3 > -(1)^2 + 5` is 3 > 4 which is false.

Point B must be true, but check to confirm: `5 > -(2)^2 + 5` is 5 > 1 which is true.

2. By drawing a graph, or otherwise, give the number of integer co-ordinates satisfied by the inequalities `y > x^2 - 4 text( and ) y < x + 1`.

Answer: 12

Draw a graph for `y = x^2 - 4 text( and ) y = x + 1`:

Choose a point bounded by both lines to ensure that the area satisfies both inequalities, say point (1, 0): `0 > 1(2) - 4` is true, and `0 < (1) + 1` is true.

The question asks for the number of integer points: because neither inequality is or equal to then count only those integer points within the area and not on the line: there are twelve of them (-1, -1), (-1, -2), (0, 0), (0, -1), (0, -2), (0, -3), (1, 1), (1, 0), (1, -1), (1, -2), (2, 2) and (2, 1).