\[y < x\]

## How to Graph Inequalities on the Coordinate Plane

- Identify whether the graph is going to have a dotted line or a solid line.
- If the inequality symbol is \(>\) or \(<\), then the line will be dotted.
- If the inequality symbol is \(\geq\) or \(\leq\), then the line will be solid.

- Graph the line as if the inequality was an equation.
- Determine which side of the line will be shaded.

## Examples

Graph the inequality:

\[y \leq x\]

*Is there a solid line or a dotted line between the shaded and unshaded parts of the graph?*

The inequality has a “less than or equal to” symbol, so there is a solid line separating the shaded and unshaded parts of the graph.

*If the \(<\) sign was an equal sign, what would the graph of the equation be?*

If the \(\geq\) symbol was an equal sign, I would have a linear equation in slope intercept form.

\[\blue y=x\]

The y intercept is 0 and the slope is 1. So, the graph of the linear equation would look like this:

*Which side of the line should be shaded?*

I’m going to test the point \(\blue (1, 3)\) to see if it is a solution to the inequality.

\[{\blue 3}\leq {\blue 1}\]

This is a false statement, so \(\blue (1,3)\) is NOT a solution of the inequality.

The non-solution \(\blue (1, 3)\) is above the line on the graph, so all of the solutions will be below the line.

This means I should shade the graph below the line.

Graph the inequality:

\[y \geq 3x-2\]

*Is there a solid line or a dotted line between the shaded and unshaded parts of the graph?*

The inequality has a “greater than or equal to” symbol, so there is a solid line separating the shaded and unshaded parts of the graph.

*If the \(\geq\) sign was an equal sign, what would the graph of the equation be?*

If the \(\geq\) symbol was an equal sign, I would have a linear equation in slope intercept form.

\[\yellow y=3x-2\]

The y intercept is 2 and the slope is 3. So, the graph of the linear equation would look like this:

*Which side of the line should be shaded?*

I’m going to test the point \(\yellow (1, 2)\) to see if it is a solution to the inequality.

\[{\yellow 2} \geq 3({\yellow 1})-2\]

\[{\yellow 2} \geq 3-2\]

\[{\yellow 2} \geq 1\]

This is a true statement, so \(\yellow (2,1)\) is a solution of the inequality.

The solution \(\yellow (2, 1)\) is above the line on the graph, so all of the other solutions will be above the line as well.

This means I should shade the graph above the line.

Graph the inequality:

\[6x+2y<-6\]

*Is there a solid line or a dotted line between the shaded and unshaded parts of the graph?*

The inequality has a “less than” symbol, so there is a dotted line separating the shaded and unshaded parts of the graph.

*If the \(<\) sign was an equal sign, what would the graph of the equation be?*

If the \(\geq\) symbol was an equal sign, I would have a linear equation in standard form.

\[\red 6x+2y=-6\]

The y intercept is -3 and the x intercept is -1. So, the graph of the linear equation would look like this:

*Which side of the line should be shaded?*

I’m going to test the point \(\red (0,0)\) to see if it is a solution to the inequality.

\[6({\red 0}) +2({\red 0})<-6\]

\[0 +0<-6\]

\[0<-6]

This is a false statement, so \(\red (0,0)\) is NOT a solution of the inequality.

The non-solution \(\red (0, 0)\) is above the line on the graph, so all of the solutions will be below the line.

This means I should shade the graph below the line.

Graph the inequality:

\[y>x^2-2x-3\]

*Is there a solid line or a dotted line between the shaded and unshaded parts of the graph?*

The inequality has a “greater than” symbol, so there is a dotted line separating the shaded and unshaded parts of the graph.

*If the \(>\) sign was an equal sign, what would the graph of the equation be?*

If the \(>\) symbol was an equal sign, I would have a parabola equation in standard form.

\[\purple y>x^2-2x-3\]

The zeros are -1 and 3. And the vertex is \((1, -4)\). So, the graph of the parabola would look like this:

*Which side of the line should be shaded?*

I’m going to test the point \(\purple (1,-3)\) to see if it is a solution to the inequality.

\[{\purple -3}>({\purple 1})^2-2( {\purple 1})-3\]

\[{\purple -3}>1-2-3\]

\[{\purple -3}>-4\]

This is a true statement, so \(\purple (1,3)\) is a solution of the inequality.

The solution \(\purple (1, 3)\) is inside the curve of the line on the graph, so all of the other solutions will also be inside the curve.

This means I should shade the graph inside the curve of the line.

## How to Write Inequalities from a Graph

## Examples

## Why It Works

## Dotted Line vs Solid Line

When you graph inequalities on a coordinate plane, part of the plane will be shaded and part of it will not. The line separating the shaded and unshaded parts will be dotted or solid depending on the inequality symbol.

- If the inequality symbol is \(>\) or \(<\), then the line will be dotted.
- If the inequality symbol is \(\geq\) or \(\leq\), then the line will be solid.

The only difference between the \(>\) and \(\geq\) symbols is that the \(\geq\) means that the two sides of the inequality can be equal to each other.

The line separating the shaded and unshaded parts of a inequalities graph is where the two sides of the inequality are equal to each toerh A dotted line means that the points on the line itself are NOT solutions to the inequality.

A solid line means that the points on the line ARE solutions to the inequality.

\[y < x\]

\[y \leq x\]