Inequalities

NZ Level 6 (NZC) Level 1 (NCEA)

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`=2`x`+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`>2`x`+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$2`x`+3, we cannot include the points on the line.

Here is another example $y\le2x+3$`y`≤2`x`+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$2`x`+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).

Select the inequalities that describe the shaded region.

Loading Graph...

$y$

`y`$\le$≤$2x-3$2`x`−3 or $y$`y`$\le$≤$1$1A$y$

`y`$\le$≤$2x-3$2`x`−3 and $y$`y`$\ge$≥$1$1B$y$

`y`$\ge$≥$2x-3$2`x`−3 and $y$`y`$\le$≤$1$1C$y$

`y`$\ge$≥$2x-3$2`x`−3 and $x$`x`$\ge$≥$1$1D$y$

`y`$\le$≤$2x-3$2`x`−3 or $y$`y`$\le$≤$1$1A$y$

`y`$\le$≤$2x-3$2`x`−3 and $y$`y`$\ge$≥$1$1B$y$

`y`$\ge$≥$2x-3$2`x`−3 and $y$`y`$\le$≤$1$1C$y$

`y`$\ge$≥$2x-3$2`x`−3 and $x$`x`$\ge$≥$1$1D

Select the inequalities that describe the shaded region.

Loading Graph...

$y$

`y`$>$>$x$`x`and $y$`y`$\le$≤$-x$−`x`A$y$

`y`$\le$≤$x$`x`and $y$`y`$>$>$-x$−`x`B$y$

`y`$>$>$x$`x`or $y$`y`$\ge$≥$-x$−`x`C$y$

`y`$<$<$x$`x`and $y$`y`$\le$≤$-x$−`x`D$y$

`y`$>$>$x$`x`and $y$`y`$\le$≤$-x$−`x`A$y$

`y`$\le$≤$x$`x`and $y$`y`$>$>$-x$−`x`B$y$

`y`$>$>$x$`x`or $y$`y`$\ge$≥$-x$−`x`C$y$

`y`$<$<$x$`x`and $y$`y`$\le$≤$-x$−`x`D

Consider the lines $y=4x-3$`y`=4`x`−3 and $y=2x-5$`y`=2`x`−5.

Find the $x$

`x`-coordinate of the point at which the two lines intersect.Hence find the $y$

`y`-coordinate of the point of intersection.Graph the region that satisifies both $y$

`y`$<$< $4x-3$4`x`−3 and $y$`y`$<$< $2x-5$2`x`−5?Loading Graph...

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

Apply knowledge of geometric representations in solving problems