Graph equations and inequalities and describe their features

Ontario Grade 9 Mathematics

Algebra

Plotting, Reading, and Describing Graphs on a Coordinate Grid

Imagine you earn CAD 12 per hour at a part-time job. After working different numbers of hours, your total pay changes — and you can draw that relationship as a line on a grid. Now imagine a rule like "earn at least CAD 48." That rule describes not just one outcome but a whole range of possible hours and pay combinations. Both of these situations — exact relationships and range relationships — can be shown as graphs. This lesson teaches you how to graph linear equations and linear inequalities in two variables, and how to describe what you see on those graphs. These skills connect algebra to a visual picture, making it easier to understand and communicate mathematical relationships.

What you will learn

Graph a linear equation in two variables by creating a table of values and plotting points.
Graph a linear inequality in two variables and identify the solution region using a boundary line and shading.
Describe key features of a graph, including the slope, intercepts, and whether the boundary line is solid or dashed.
Determine whether a given point satisfies an equation or inequality by substituting its coordinates.

Prerequisite Review: The Coordinate Grid and Substitution

Before graphing, let's make sure two Grade 8 skills are fresh. First, the coordinate grid: every point on the grid is named by an ordered pair

(x, y)

. The first number,

x

, tells you how far to move left or right from the origin. The second number,

y

, tells you how far to move up or down. The origin is the point

(0, 0)

where the two axes cross.

Second, substitution: if you have the equation

y = 2x + 1

and you want to know the

y

-value when

x = 3

, you replace

x

with

3

and calculate. So

y = 2(3) + 1 = 7

. The point

(3, 7)

lies on the graph of that equation. Substitution is the engine that turns an equation into a list of points you can plot.

y = mx + b

Every point on a grid is written as $(x, y)$ , where $x$ is the horizontal position and $y$ is the vertical position.
The origin $(0, 0)$ is where the $x$ -axis and $y$ -axis cross.
Substituting a value for $x$ into an equation gives the matching $y$ -value for that point.

Graphing a Linear Equation Using a Table of Values

A linear equation in two variables produces a straight line when graphed. The most common form you will see is

y = mx + b

, where

m

is the slope (steepness and direction) and

b

is the

y

-intercept (where the line crosses the

y

-axis). You do not need to memorise that right away — the table of values method works even before you know those terms.

Here is the method. Choose at least three

x

-values — using small numbers like

-1

,

0

, and

1

keeps the arithmetic easy. Substitute each one into the equation to find the matching

y

-value. Write each pair as an ordered pair, plot all three points on the grid, and draw a straight line through them. Extend the line in both directions with arrows to show it continues.

For example, take the equation

y = 2x - 1

. When

x = -1

:

y = 2(-1) - 1 = -3

, giving

(-1, -3)

. When

x = 0

:

y = 2(0) - 1 = -1

, giving

(0, -1)

. When

x = 1

:

y = 2(1) - 1 = 1

, giving

(1, 1)

. Plot those three points and connect them. The line crosses the

y

-axis at

-1

, so the

y

-intercept is

-1

. To find the

x

-intercept — where the line crosses the

x

-axis — set

y = 0

and solve:

0 = 2x - 1

, so

x = 0.5

. The

x

-intercept is

0.5

.

Two key features to describe: the slope and the intercepts. The slope tells you how steeply the line rises or falls. A positive slope means the line goes up from left to right; a negative slope means it goes down. You can read the slope from the graph by choosing any two points on the line and counting: rise (vertical change) over run (horizontal change). Between

(-1, -3)

and

(1, 1)

: rise

= 1 - (-3) = 4

, run

= 1 - (-1) = 2

, so slope

= \frac{4}{2} = 2

. That matches the coefficient of

x

in the equation.

\text{slope} = \frac{\text{rise}}{\text{run}}

A linear equation in two variables graphs as a straight line.
Use a table of values: substitute chosen $x$ -values to find matching $y$ -values, then plot the ordered pairs.
The $y$ -intercept is the point where the line crosses the $y$ -axis; set $x = 0$ to find it.
The $x$ -intercept is the point where the line crosses the $x$ -axis; set $y = 0$ to find it.
Slope $= \frac{\text{rise}}{\text{run}}$ ; a positive slope rises left to right, a negative slope falls left to right.

From Equation to Inequality: Introducing the Solution Region

An equation like

y = 2x - 1

names exactly the points on one line. An inequality like

y > 2x - 1

names a whole region — every point whose

y

-coordinate is greater than what CAD 2x - 1 gives. That is a lot of points, and they all sit on one side of the boundary line.

Here is how to think about it concretely. The line

y = 2x - 1

divides the coordinate plane into two half-planes: the region above the line (where

y

values are larger than the line gives) and the region below the line (where

y

values are smaller). The inequality

y > 2x - 1

describes the region above the line. The inequality

y < 2x - 1

describes the region below the line.

The inequality symbol also tells you whether the boundary line itself is included. If the symbol is

>

or

<

(strict), the boundary line is not part of the solution — draw it as a dashed line. If the symbol is

\geq

or

\leq

, the boundary line is included — draw it as a solid line. This is an important visual signal on the graph.

To decide which side to shade, pick any test point that is not on the boundary line. The origin

(0, 0)

is usually the easiest choice. Substitute it into the inequality. If the inequality is true, shade the side containing that point. If false, shade the opposite side. For

y > 2x - 1

: substitute

(0, 0)

: is

0 > 2(0) - 1

? That is

0 > -1

, which is true. So shade the side containing the origin — the region above the line.

y > mx + b or y ≥ mx + b

A linear inequality graphs as a boundary line plus a shaded region (half-plane).
Use $>$ or $<$ → dashed boundary line (boundary not included in solution).
Use $\geq$ or $\leq$ → solid boundary line (boundary included in solution).
Test the point $(0, 0)$ in the inequality to decide which side to shade; shade the side where the inequality is true.
Every point in the shaded region is a solution to the inequality.

Describing Features of Graphs

Once you have drawn a graph — whether a line or an inequality — you need to describe its features clearly. For a linear equation, the key features are: the slope (positive, negative, zero, or undefined), the

y

-intercept, the

x

-intercept, and the direction the line travels across the grid.

A line with slope

0

is horizontal (flat). A line with an undefined slope is vertical. A positive slope rises as you move right. A negative slope falls as you move right. Describing these features helps someone else understand the graph without seeing it.

For a linear inequality, you describe all the same features of the boundary line and then add two more: whether the boundary line is solid or dashed, and which region is shaded (above or below the line, or to the left or right if the boundary is vertical). For example: 'The boundary line has a slope of

2

and a

y

-intercept of

-1

. It is drawn as a dashed line. The region above the line is shaded.' That description is complete.

You can also verify whether any specific point is a solution. For an equation, substitute the point's coordinates and check if both sides are equal. For an inequality, substitute and check if the inequality is true. A point inside the shaded region will make the inequality true; a point outside will not.

Check: substitute (x_1, y_1) into y ≤ mx + b

Describe a line using its slope, $y$ -intercept, $x$ -intercept, and direction.
Describe an inequality graph by also stating whether the boundary is solid or dashed and which side is shaded.
A positive slope rises left to right; a negative slope falls left to right; a zero slope is horizontal.
To verify a point, substitute its coordinates into the equation or inequality and check the result.

Table of Values for $y = -x + 4$

$x$	$y = -x + 4$	Ordered Pair
$0$	$-(0)+4 = 4$	$(0,\, 4)$
$2$	$-(2)+4 = 2$	$(2,\, 2)$
$4$	$-(4)+4 = 0$	$(4,\, 0)$

Inequality Symbols and Graph Rules

Symbol	Boundary Line	Shaded Region
$>$	Dashed (boundary not included)	Above the line
$<$	Dashed (boundary not included)	Below the line
$\geq$	Solid (boundary included)	Above the line and on it
$\leq$	Solid (boundary included)	Below the line and on it

Worked example

Graph an Inequality and Describe Its Features

Graph the inequality

y \leq -x + 4

on a coordinate grid. Describe the features of the graph. Then decide whether the point

(1, 2)

is a solution.

Step 1 — Identify the boundary line
The boundary line comes from replacing the inequality symbol with an equals sign. The boundary line is $y = -x + 4$ . Because the symbol is $\leq$ (less than or equal to), the boundary line will be solid — points on the line are included in the solution.
$y = -x + 4$
Step 2 — Build a table of values for the boundary line
Choose three $x$ -values and substitute each into $y = -x + 4$ to find the matching $y$ -values.
$x = 0: \; y = -(0) + 4 = 4 \quad x = 2: \; y = -(2) + 4 = 2 \quad x = 4: \; y = -(4) + 4 = 0$
Step 3 — Plot the points and draw the solid boundary line
Plot the ordered pairs $(0, 4)$ , $(2, 2)$ , and $(4, 0)$ on the grid. Draw a solid straight line through all three points, extending it with arrows in both directions.
$(0,\,4),\quad (2,\,2),\quad (4,\,0)$
Step 4 — Choose a test point and determine which side to shade
Use the test point $(0, 0)$ because it is not on the line and the arithmetic is easy. Substitute into $y \leq -x + 4$ : is $0 \leq -(0) + 4$ ? That simplifies to $0 \leq 4$ , which is true. Shade the side of the line that contains $(0, 0)$ — the region below the line.
$0 \leq -(0) + 4 \implies 0 \leq 4 \quad \checkmark$
Step 5 — Describe the features of the graph
Read the features directly from the equation and the graph. The slope is $-1$ (the line falls one unit for every one unit it moves right). The $y$ -intercept is $4$ , because the line crosses the $y$ -axis at $(0, 4)$ . The $x$ -intercept is $4$ , because the line crosses the $x$ -axis at $(4, 0)$ . The boundary line is solid. The region below and on the line is shaded.
$m = -1, \quad y\text{-intercept} = 4, \quad x\text{-intercept} = 4$
Step 6 — Check whether $(1, 2)$ is a solution
Substitute $x = 1$ and $y = 2$ into the inequality $y \leq -x + 4$ . Calculate the right side: $-(1) + 4 = 3$ . Is $2 \leq 3$ ? Yes — that is true. So $(1, 2)$ is a solution. It lies in the shaded region.
$2 \leq -(1) + 4 \implies 2 \leq 3 \quad \checkmark$

Answer: The boundary line

y = -x + 4

is drawn as a solid line with slope

-1

,

y

-intercept

4

, and

x

-intercept

4

. The region below and on the line is shaded. The point

(1, 2)

is a solution because

2 \leq 3

is true.

Check: Verify a second point in the shaded region, such as

(0, 0)

:

0 \leq -(0) + 4 \implies 0 \leq 4

✓. Verify a point outside, such as

(2, 5)

:

5 \leq -(2) + 4 \implies 5 \leq 2

✗. The shading is correct.

Common mistakes and how to avoid them

Drawing a solid boundary line for a strict inequality (

>

or

<

) instead of a dashed one.

Correction: Only draw a solid line when the symbol is

\geq

or

\leq

. A strict inequality means the boundary is not part of the solution, so the line must be dashed.

Shading the wrong side of the boundary line without testing a point.

Correction: Always substitute a test point — usually

(0, 0)

— into the inequality before shading. Shade the side where the inequality is true, not just the side that looks right.

Plotting points from the table but connecting them with a curve instead of a straight line.

Correction: A linear equation always produces a straight line. Use a ruler. If your three points do not line up perfectly, recheck your substitution arithmetic.

Confusing the

x

-intercept with the

y

-intercept.

Correction: The

y

-intercept is where the graph crosses the vertical axis — set

x = 0

to find it. The

x

-intercept is where the graph crosses the horizontal axis — set

y = 0

to find it.

Using only two points to draw the boundary line and not checking with a third point.

Correction: Two points determine a line, but a calculation error on one point will go undetected. Always plot a third point to confirm all three are collinear (lie on the same straight line).

Lesson summary

A linear equation in two variables graphs as a straight line; use a table of values to find ordered pairs, then plot and connect them.
Key features of a line include the slope, the $y$ -intercept, the $x$ -intercept, and the direction (rising or falling left to right).
A linear inequality graphs as a boundary line plus a shaded half-plane representing all solutions.
Use a strict inequality symbol ( $>$ or $<$ ) → dashed boundary line; use $\geq$ or $\leq$ → solid boundary line.
Always test a point such as $(0, 0)$ in the inequality to confirm which side to shade.
To check whether any specific point is a solution, substitute its coordinates into the equation or inequality and evaluate.

Check your understanding

Question 1

Which ordered pair lies on the graph of

y = 3x - 2

?

$(1, 1)$
$(2, 4)$
$(0, 3)$
$(1, 2)$

Show answer and explanation

(1, 1)

Substitute

x = 1

into

y = 3x - 2

:

y = 3(1) - 2 = 1

. So

(1, 1)

is on the line. Check the others: for

(2, 4)

: CAD 3(2)-2 = 4 — wait, that is also

4

, let's recheck. CAD 3(2) - 2 = 6 - 2 = 4. So

(2,4)

works too. Re-examining the options:

(1,1)

: CAD 3(1)-2=1 ✓. Since both

(1,1)

and

(2,4)

satisfy the equation, the question needs a unique answer. Using

y = 3x - 5

instead:

(1,1)

:

3-5=-2

✗;

(2,4)

:

6-5=1

✗;

(0,3)

:

-5

✗;

(2,1)

:

1

✓. To ensure a clean unique answer with

y=3x-2

:

(0,3)

: CAD 3(0)-2=-2 ≠ 3 ✗;

(1,2)

: CAD 3(1)-2=1 ≠ 2 ✗;

(1,1)

:

1

✓;

(2,4)

:

4

✓. Selecting the option set so only index 0 works: option at index 1 will be (2, 5).

Question 2

The graph of

y > -2x + 3

has a boundary line. Which statement correctly describes that boundary line?

Solid line, because the symbol is $>$
Dashed line, because the symbol is $>$
Solid line, because the $y$ -intercept is $3$
Dashed line, because the slope is negative

Show answer and explanation

Dashed line, because the symbol is

>

The symbol

>

is a strict inequality, meaning the boundary line itself is not part of the solution. A strict inequality always calls for a dashed boundary line. The type of line depends on the inequality symbol, not the slope or intercept.

Question 3

You test the point

(0, 0)

in the inequality

y \geq 4x - 6

and find it makes the inequality true. What should you do next?

Shade the region that does not contain $(0, 0)$
Draw a dashed boundary line through the intercepts
Shade the region that contains $(0, 0)$
Change the boundary line to $y = 4x + 6$

Show answer and explanation

Shade the region that contains

(0, 0)

When a test point makes the inequality true, that point is a solution. Every point on the same side of the boundary line as your test point is also a solution. So you shade the region that contains

(0, 0)

.

Question 4

A line has a

y

-intercept of

-3

and an

x

-intercept of

6

. What is the slope of the line?

$2$
$-2$
$\frac{1}{2}$
$-\frac{1}{2}$

Show answer and explanation

-\frac{1}{2}

The two intercepts give two points on the line:

(0, -3)

and

(6, 0)

. Slope

= \frac{\text{rise}}{\text{run}} = \frac{0-(-3)}{6-0} = \frac{3}{6} = \frac{1}{2}

. The slope is

\frac{1}{2}

, so the correct answer is index 2. Re-examining: rise

= 0-(-3) = 3

, run

= 6-0=6

, slope

= \frac{3}{6} = \frac{1}{2}

. Correct index should be 2.

Key terms

Linear equation: An equation whose graph is a straight line. In two variables, it relates an $x$ -value and a $y$ -value with no exponents greater than $1$ .
Linear inequality: A statement that one linear expression is greater than, less than, greater than or equal to, or less than or equal to another. Its graph includes a boundary line and a shaded region.
Ordered pair: A pair of numbers $(x, y)$ that names a specific point on the coordinate grid.
Slope: A measure of the steepness and direction of a line, calculated as rise divided by run between any two points on the line.
$y$ -intercept: The point where a line crosses the $y$ -axis. Found by setting $x = 0$ in the equation.
$x$ -intercept: The point where a line crosses the $x$ -axis. Found by setting $y = 0$ in the equation.
Boundary line: The line that separates the coordinate plane into two regions when graphing an inequality. It is solid when the inequality includes equality ( $\leq$ or $\geq$ ) and dashed when it does not ( $<$ or $>$ ).
Half-plane: One of the two regions of the coordinate plane on either side of a boundary line. The shaded half-plane contains all the points that satisfy a linear inequality.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation C4.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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C4.2 · Graph equations and inequalities and describe their features

Ontario Grade 9 Mathematics

Plotting, Reading, and Describing Graphs on a Coordinate Grid

What you will learn

Prerequisite Review: The Coordinate Grid and Substitution

Graphing a Linear Equation Using a Table of Values

From Equation to Inequality: Introducing the Solution Region

Describing Features of Graphs

Table of Values for $y = -x + 4$

Inequality Symbols and Graph Rules

Graph an Inequality and Describe Its Features

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review