Representing inequalities graphicallyEdexcel International A Level Maths: Revision notes

The inequality $y>x+r$ describes every point above the line $y=x+r$. The inequality $y<x+r$ describes every point below it. The line itself is the boundary. A region is drawn by sketching the boundary, deciding between dotted and solid, then shading the correct side. Shade the region and label it $R$ so it is clear which side is wanted. For vertical boundaries, $x>a$ is the region to the right of $x=a$ and $x<a$ is to the left; horizontal boundaries $y>b$ and $y<b$ are above and below.

Dotted (broken) line: the inequality is strict, $<$ or $>$, so points on the boundary are not in the region. Solid line: the inequality is $\le$ or $\ge$, so points on the boundary are in the region. For $y\le5$ the line $y=5$ is solid. For $y>x+1$ the line $y=x+1$ is dotted. Where a solid line meets a dotted line, the meeting point is excluded, because it lies on the dotted boundary.

For $y>ax^2+bx+c$, sketch the parabola first: find the $x$-intercepts (solve $ax^2+bx+c=0$), the $y$-intercept ($c$) and the turning point, e.g. by completing the square. For $y=x^2-2x-3$: roots $-1$ and $3$, $y$-intercept $-3$, minimum $(1,-4)$. Then $y>x^2-2x-3$ is the region above the curve (the points 'inside' the U shape), drawn with a dotted curve. For $y<x^2-2x-3$, shade below the curve (outside the U shape).

If you are unsure which side to shade, choose a point that is not on the boundary (the origin is easiest if it is not on it) and substitute. Example: for $y>x^2-2x-3$ test $(0,0)$: $0>-3$ is true, so shade the side containing the origin. If the statement is false, shade the other side.

When a region satisfies two inequalities at once, shade only the part that satisfies both. Where two boundaries meet, solve them simultaneously, e.g. $x^2-6x+5=x-1$ gives $x=1$ and $x=6$ and the points $(1,0)$ and $(6,5)$. The region $y>x^2-6x+5$ and $y\le x-1$ lies between the dotted curve and the solid line for $1<x<6$. Points on the line between the intersections are in the region, but the two intersection points themselves are not, because they lie on the dotted curve.