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## Digital SAT Math

### Unit 2: Lesson 8

Graphs of linear systems and inequalities: foundations# Graphs of linear systems and inequalities | Lesson

A guide to graphs of linear systems and inequalities on the digital SAT

## What are graphs of linear systems and inequalities problems?

Graphs of linear systems and inequality problems deal with graphs of the following in the x, y-plane:

In this lesson, we'll learn to:

- Identify solutions to linear systems and inequalities
- Match graphs with systems of linear equations and inequalities

This lesson builds upon an understanding of the following skills:

- Solving linear equations and linear inequalities
- Graphs of linear equations and functions
- Solving systems of linear equations

**You can learn anything. Let's do this!**

## How are systems of linear equations related to their graphs?

### Systems of equations with graphing

### Intersection points as solutions

In a system of linear equations, each equation is represented by a line in the x, y-plane. The intersection of these lines represents the solution to the system.

For example, the graphs of y, equals, x and y, equals, minus, x intersect at left parenthesis, 0, comma, 0, right parenthesis, which is the solution to the system of equations.

### Try it!

## How do I graph linear inequalities?

### Intro to graphing two-variable inequalities

### Representing solutions with shaded areas

When we learned to solve linear inequalities, we learned that multiple values can satisfy an inequality. How do we graph the infinite number of left parenthesis, x, comma, y, right parenthesis values that can satisfy a linear inequality?

We do it by shading the x, y-plane! A linear inequality is defined by a line in the x, y-plane. The line divides the plane into two halves, and which half of the plane we shade depends on the inequality sign.

It's the easier to determine which half of the plane to shade when the inequality is in slope-intercept form. When we have linear inequalities in slope-intercept form:

- If y is greater than m, x, plus, b, shade above the line.
- If y is less than m, x, plus, b, shade below the line.

But what about points

*on*the line? In slope-intercept form, inequalities with...- Greater than (is greater than) or less than (is less than) signs do
*not*include points on the line in the solution set. We use a**dashed line**to show that the points on the line are not included. - Greater than or equal to (is greater than or equal to) or less than or equal to (is less than or equal to) signs do include points on the line in the solution set. We use a
**solid line**to show that points on the line are included.

For example, for start color #7854ab, y, is greater than, x, end color #7854ab, we draw y, equals, x as a

*dashed*line and shade*above*the line:### Try it!

## How do I graph systems of linear inequalities?

### Intro to graphing systems of inequalities

### How do I identify the region representing a system of linear inequalities?

Two intersecting lines will always divide the x, y-plane into four regions. Points in each of the four regions represent solutions to a different system of linear inequalities.

In the graph below, the equations of the lines defining the four regions are:

Replacing the equal signs with inequality signs lets us specify one of the four regions.

When we have linear inequalities in slope-intercept form:

- If y is greater than m, x, plus, b, shade above the line.
- If y is less than m, x, plus, b, shade below the line.

For example, let's look at the following system:

This means the solutions to the system of linear inequalities are represented by the region

*below*the line y, equals, 3, x, minus, 2 and*above*the line y, equals, start fraction, 1, divided by, 3, end fraction, x, plus, start fraction, 4, divided by, 3, end fraction:### Try it!

## Your turn!

## Things to remember

When a system of linear equations is graphed, the solution appears where the lines intersect.

When we have linear inequalities in slope-intercept form:

- If y is greater than m, x, plus, b, shade above the line.
- If y is less than m, x, plus, b, shade below the line.

In slope-intercept form, inequalities with...

- Greater than (is greater than) or less than (is less than) signs do
*not*include points on the line in the solution set. We use a**dashed line**to show that the points on the line are not included. - Greater than or equal to (is greater than or equal to) or less than or equal to (is less than or equal to) signs do include points on the line in the solution set. We use a
**solid line**to show that points on the line are included.

## Want to join the conversation?

- I couldn't understand the practice: intersection of two lines(4 votes)
- The example for PRACTICE: MATCH A SYSTEM OF LINEAR INEQUALITIES TO ITS GRAPH is wrong.(0 votes)
- It is not. The answer is D, because at the end of separating y in x-y≥4 you multiply the inequality by -1, thus changing the direction of the inequality symbol, and 4 into -4: y≤x-4(3 votes)