03Ginequations

toc = Sketching Inequations =

Recall that the graph of a function like ** y = 2x + 3 ** produces a straight line. Any point on that line makes the equation true. {For example, the point ** (0, 3) ** → ** 3 = 0 + 3 ** (true) }

In the same way, the graph of a function like ** y > 2x + 3 ** will produce a region (or area). Any point in that region will make the inequation true. {For example, the point ** (–1, 4) ** **→** ** 4 > –2 + 3 ** (true) }

Shading and Lines
To indicate the correct region, we shade out the part of the graph __not__ required.

The straight line, ** y = 2x + 3 ** will provide the boundary (or edge) of the region.

For inequations using ** < ** or ** > **, we use a __dashed line__ to indicate that the region does __not__ include the boundary. {ie the points on the boundary do __not__ make the inequation true}

For inequations using ** __<__ ** or ** __>__ **, we use a __solid line__ to indicate that the region __does__ include the boundary. {ie the points on the boundary __do__ make the inequation true}

To decide where to shade:
 * Select a point that is clearly not on the boundary. (Hint: select a point with easy numbers)
 * Substitute that point into the inequation.
 * if the result is __true__, the point __is__ in the region required
 * so shade the __opposite side__ of the boundary from the selected point
 * if the result is __false__, the point is __not__ in the region required
 * so shade the __same side__ of the boundary as the selected point

{For example, for ** y > 2x + 3 **, ** (–1, 4) ** gives __true__, so the __opposite__ side of the boundary is shaded}

Inequality Signs
Many text books and web sites (including this one, sometimes) show This is due to limitations in the character set available. The second line should be sloping and not horizontal.
 * the " less than or equal to " sign as __<__
 * the " greater than or equal to " sign as __>__

math \text{You should always write } \leqslant \text{ and not } \underline{<} math math \text{Similarly, you should always write } \geqslant \text{ and not } \underline{>} math

** Example 1 **
Sketch y < 3x – 2

__**Solution:**__

First graph the boundary line: ** y = 3x – 2 **
 * ** c = –2 ** so y-intercept at ** (0, –2) **
 * ** m = 3 **so rise = 3, run = 1
 * so 2nd point at ** (1, 1) **
 * inequality sign is ** < ** so use __broken line__.

To decide where to shade, select a point not on the boundary.
 * ** (0, 0) ** is selected, so substitute into inequation:
 * ** 0 < 0 – 2 **: this is NOT true
 * so ** (0, 0) ** is NOT in the region required
 * we shade out the region NOT required
 * so shade the area including ** (0, 0) **

** Example 2 **
math \text{Sketch } 2x + 4y \leqslant 8 math __**Solution:**__

First graph the boundary line: ** 2x + 4y = 8 **
 * ** x-intercept : **let y = 0
 * 2x + 0 = 8
 * gives x = 4 so ** (4, 0) **
 * ** y-intercept : **let x = 0
 * 0 + 4y = 8
 * gives y = 2 so ** (0, 2) **
 * inequality sign is ** __<__ ** so use __solid line__

To decide where to shade, select a point not on the boundary Go to top of page flat .
 * ** (0, 0) ** is selected, so substitute into inequation
 * ** 0 + 0 __<__ 8 ** IS true
 * so ** (0, 0) ** IS in the required region
 * so shade the area that does __not__ include ** (0, 0) **