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.

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

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

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

Here is the inequality y > 2x+3.

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

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

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

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

Here is another example y\leq 2x + 3.

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

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

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

Examples

Example 1

Select the inequalities that describe the shaded region.

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x+y \leq 9, x\geq 3

x+y > 9, x< 3

x+y > 9, x> 3

x+y \leq 9, x\leq 3

Worked Solution

Create a strategy

If the lines are dotted and the shaded region is above or on the right of each line then the inequality symbol is >.

Apply the idea

The answer is option C.

Example 2

Select the inequalities that describe the shaded region.

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y \geq x and y \geq -x

y \geq x and y \leq -x

y \leq x and y \geq -x

y \leq x and y \leq -x

Worked Solution

Create a strategy

If the lines are solid and the shaded region is above each line then the inequality symbol is \geq.

Apply the idea

The answer is option A.

Example 3

Consider the lines y=4x-2 and y=3x-3.

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

Worked Solution

Create a strategy

Equate each left-hand side of the equations.

Apply the idea

\displaystyle 4x-2	\displaystyle =	\displaystyle 3x-3	Equate each left-hand side
\displaystyle 4x-2-3x	\displaystyle =	\displaystyle 3x-3-3x	Subtract 3x from both sides
\displaystyle x-2	\displaystyle =	\displaystyle -3	Evaluate
\displaystyle x-2+2	\displaystyle =	\displaystyle -3+2	Add 2 to both sides
\displaystyle x	\displaystyle =	\displaystyle -1	Evaluate

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

Worked Solution

Create a strategy

Substitute x=-1 into one of the equations.

Apply the idea

We can use y=4x-2.

\displaystyle y	\displaystyle =	\displaystyle 4x-2	Write the equation
	\displaystyle =	\displaystyle 4\times(-1)-2	Substitute x=-1
	\displaystyle =	\displaystyle -4-2	Evaluate the multiplication
	\displaystyle =	\displaystyle -6	Evaluate

Graph the region that satisifies both y<4x-2 and y<3x-3.

Worked Solution

Create a strategy

Plot the intersection point and the y-intercepts of each inequality.
The inequality symbols are both < so the lines must be dotted.
Since it is an "and" inequality then the shaded region must be located to the values of the inequalities.

Apply the idea

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We found from parts (a) and (b) that the intersection point is at (-1,-6).

y<4x-2 has a y-intercept at (0,-2), while y<3x-3 has a y-intercept at (0,-3).

Since y<4x-2 (green dotted line) is above y<3x-3 (blue dotted line) then the shaded region must below y<3x-3 since it satisfies the values of both inequalites.

Idea summary

The following are features of two variable inequalities.

A line is solid if the symbols are \geq or \leq.
A line is dotted if the symbols are > or <.
If the symbols are \geq and >, then the shaded region must be above the line.
If the symbols are \leq and <, then the shaded region must be below the line.
An "and" inequality shows a shaded region located to the values of both inequalities.

Outcomes

AC9M10A02

solve linear inequalities and simultaneous linear equations in 2 variables; interpret solutions graphically and communicate solutions in terms of the situation

3.03 Graphs of inequalities in two variables

Ideas

Graphs of two variable inequalities

Examples

Example 1

Example 2

Example 3

Outcomes

AC9M10A02

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