New Zealand
Level 6 - NCEA Level 1

# Regions in the Number Plane

Lesson

As we have seen in our work with inequalities (see these entries to remind yourself if you need), an inequality states a range of solutions to a problem instead of just a singular answer.

The difference is best described with an example:

Here is the line $y=2x+3$y=2x+3

The line shows all the solutions to the equation.  All the possible $y$y values that make this equation true for any $x$x value that is chosen.

For every $x$x value there is only one possible corresponding $y$y value.

For example, if $x=1$x=1, then according to the equation $y=5$y=5 (as marked on the diagram)

Here is the inequality $y>2x+3$y>2x+3

The solution to this is not a single line, as for every $x$x value, there are multiple $y$y values that satisfy the inequality.  The solution graph is therefore a region.

A coloured in space indicating all the possible coordinates $\left(x,y\right)$(x,y) that satisfy the inequality.

For example, at $x=1$x=1,  $y>5$y>5. So any coordinate with an $x$x value of $1$1 and a $y$y value larger than $5$5 is a solution.

The dotted line corresponds to the strictly greater than symbol that was used. That is, since $y$y cannot equal $2x+3$2x+3, we cannot include the points on the line.

Here is another example $y\le2x+3$y2x+3

Again we have a region, and this time we also have solid line indicating that the $y$y value can be less than or EQUAL to $2x+3$2x+3, for any given $x$x.

For example, if we choose $x=3$x=3, the points that satisfy the inequality are all the points whose $y$y value is less than or equal to $2\times3+3$2×3+3 or $9$9.

There are many points that do this. One such point would be $\left(3,8\right)$(3,8).

#### Examples

##### Question 2

Select the inequalities that describe the shaded region.

1. $y$y$>$>$x$x and $y$y$\le$$-xx A yy\le$$x$x and $y$y$>$>$-x$x

B

$y$y$>$>$x$x or $y$y$\ge$$-xx C yy<<xx and yy\le$$-x$x

D

$y$y$>$>$x$x and $y$y$\le$$-xx A yy\le$$x$x and $y$y$>$>$-x$x

B

$y$y$>$>$x$x or $y$y$\ge$$-xx C yy<<xx and yy\le$$-x$x

D

##### Question 3

Consider the lines $y=4x-3$y=4x3 and $y=2x-5$y=2x5.

1. Find the $x$x-coordinate of the point at which the two lines intersect.

2. Hence find the $y$y-coordinate of the point of intersection.

3. Graph the region that satisifies both $y$y $<$< $4x-3$4x3 and $y$y $<$< $2x-5$2x5?

### Outcomes

#### GM6-7

Use a co-ordinate plane or map to show points in common and areas contained by two or more loci

#### 91033

Apply knowledge of geometric representations in solving problems