Graph inequalities on a coordinate plane

Graphing an inequality means showing every point that belongs to its solution set. Rather than producing one numerical answer, you draw a boundary line and shade the side where the rule is true. The shaded region represents all the values that satisfy the inequality.

The coordinate plane has an (x)-axis, a (y)-axis and an origin at ((0,0)). Each ordered pair identifies a point, while the signs of its coordinates determine its position. This matches the orientation of Australian city maps: east is right and north is up, whether you are following Melbourne’s rail network or reading a street plan of Canberra.

An inequality uses a sign such as (<), (>), (\leq) or (\geq) to compare expressions. Its solution set may extend across an entire half-plane rather than stopping at one point. Australian students often call the shading process “shade it in”, but the important idea is deciding which side contains valid ordered pairs.

CurryPlum combines clear explanations with online homework assistance, calculators and other maths tools for revision. Australian senior-secondary students are generally familiar with CAS calculators, although graph paper and a pencil are enough for this topic. The same reasoning remains important in VCE, HSC and other upper-secondary courses that assess both method and accuracy.

Read the inequality before drawing

An equation describes equality, so (y=2x+1) identifies a single boundary line. An inequality allows a range of values, so (y>2x+1) identifies every point on one side of that line. The comparison sign tells you whether the boundary itself belongs to the solution set.

Before using coordinates, isolate the (y)-term where possible. For example, (2y\leq 4x+6) becomes (y\leq 2x+3). This form makes the boundary easy to graph: plot ((0,3)), plot ((1,5)), join them and extend the line. The inequality still determines which side receives the shading.

Set the boundary line

Replace the comparison sign with an equals sign to find the boundary. Use a solid line for (\leq) or (\geq), because the boundary points are included. Use a dashed line for (<) or (>), because those boundary points are excluded.

Next, choose a test point and substitute its coordinates into the original inequality. The origin is often convenient, but if it lies on the boundary, choose another point such as ((0,1)). A true statement means the test point belongs in the shaded region.

Regular practice turns a rule into a reliable habit, much like the patterns developed through teaching young monks. Repeating the same sequence—find the boundary, test a point, then shade—helps prevent small sign errors.

Shade the correct half-plane

The shaded region is a half-plane because every point on one side of the boundary satisfies the inequality. For (y>2x+1), testing ((0,0)) gives (0>1), which is false. Since the origin lies below the line, shade below (y=2x+1).

Reversing the comparison reverses the shading. For (y<2x+1), the origin satisfies the inequality, so shade above the boundary. The line stays dashed in both examples because equality is not allowed. Checking one valid and one invalid point confirms that the shading has not been placed on the wrong side.

Graph vertical and horizontal boundaries

Some inequalities naturally produce a vertical or horizontal line. For (x>3), draw a dashed line at (x=3) and shade to the right. For (y\leq-1), draw a solid line at (y=-1) and shade below it.

A negative coefficient requires particular care. Dividing both sides of (-2x>6) by (-2) reverses the sign, giving (x<-3). Draw a dashed vertical line at (x=-3), then shade to the left. This sign reversal is a frequent source of mistakes in Australian maths exercises and homework help requests.

Verify points and solve compound rules

A completed graph should make every solution visible. Plot several ordered pairs, including points on both sides of the boundary, and substitute them into the original inequality. The intercepts—where the boundary meets each axis—are also useful reference points.

Two inequalities joined by “and” require points to satisfy both rules, so shade the overlapping or intersection region. When they are joined by “or”, shade both possible regions. For example, (x<3) and (y>-2) create an intersection above (y=-2) and to the left of (x=3).

A cutoff can also describe the spread of numerical data. CurryPlum explains calculating standard deviation to show how values are distributed around a mean, providing a useful connection between inequalities and data analysis.

A quick sign and boundary checklist

These rules determine the boundary and its appearance:

• Replace the inequality sign with equals to locate the boundary.
• Draw a solid line when the sign includes equality.
• Draw a dashed line when the sign excludes equality.
• Rewrite the rule as (y=\text{expression}) when the line is not already in that form.

These checks protect the completed graph:

• Use a test point that does not sit directly on the boundary.
• Check the slope so the line has the correct direction.
• Plot one point from the shaded region and one from the opposite side.
• Substitute both points into the original inequality to verify the result.

If a check fails, work backwards rather than changing the shading repeatedly. Re-isolate the variable, redraw the boundary and repeat the test-point step. This approach distinguishes an algebra error from a simple shading error.

Work through a mixed example

Begin with (y\leq -x+2). Its boundary is (y=-x+2), and the solid line passes through ((0,2)) and ((2,0)). The origin satisfies (0\leq2), so shade the half-plane below the line.

Now graph (x<3) together with (y>-2). Draw a dashed vertical line at (x=3) and a dashed horizontal line at (y=-2). Because the inequalities are joined by “and”, shade only the upper-left region where both conditions hold.

The next exercise is to graph (2x-3y\geq6) on coordinate paper: draw the solid boundary (y=\frac23x-2), test the origin, and shade the half-plane containing it.