lesson 2.10 solving linear inequalities in two variables concept: represent and solve systems of...

Lesson 2.10

Solving Linear Inequalities in Two Variables

Concept: Represent and Solve Systems of Inequalities Graphically

EQ: How do I represent the solutions of an inequality in two variables? (Standard REI.12)

Vocabulary: Solutions region, Boundary lines (dashed or solid), Inclusive, Non-inclusive, Half plane, Test Point

2.3.1: Solving Linear Inequalities in Two Variables

1

IntroductionSolving a linear inequality in two variables is similar to graphing a linear equation, with a few extra steps that will be explained on the following slides.

*Remember that inequalities have infinitely many solutions and all the solutions get represented through the use of shading.

http://youtu.be/Eiwi3FvQumU

2

2.3.1: Solving Linear Inequalities in Two Variables

Key Concepts• A linear inequality in two variables has a half plane as

the set of solutions.

• A half plane is a region containing all points that has one boundary, which is a straight line that continues in both directions infinitely.

3

2.3.1: Solving Linear Inequalities in Two Variables

Inequality Brothers

4

2.3.1: Solving Linear Inequalities in Two Variables

> < ≥ ≤

Non-Inclusiveaka: Open

Inclusiveaka: Closed

5

Steps to Graphing a Linear Inequality in Two Variables

1. Graph the inequality as a linear equation. * If the inequality is inclusive (≤ and ≥), use a solid line. * If the inequality is non-inclusive (< and >), use a dashed line.2. Shade the half plane above the y-intercept for (> and ≥). Shade the half plane below the y-intercept for (< and ≤).3. Check your answer by picking a test point and substituting it into the inequality:• A test point in the shaded half plane should give you a true

statement. • A test point in the non-shaded half plane should give you a

false statement.

2.3.1: Solving Linear Inequalities in Two Variables

Guided Practice - Example 1 Graph the solutions to the following inequality.

6

2.3.1: Solving Linear Inequalities in Two Variables

Guided Practice: Example 1, continued

1. Graph the inequality as a linear equation. Since the inequality is non-inclusive, use a dashed line.y = x + 3

To graph the line, plot the y-intercept first, (0, 3). Then use the slope to find a second point. The slope is 1. Count up one unit and to the right one unit and plot a second point. Connect the two points and extend the line to the edges of the coordinate plane.

7

2.3.1: Solving Linear Inequalities in Two Variables

2.3.1: Solving Linear Inequalities in Two Variables

8

Guided Practice: Example 1, continued

2. Shade the appropriate area. Since the symbol > is used we will shade above the y-intercept.

2.3.1: Solving Linear Inequalities in Two Variables

9

Guided Practice: Example 1, continued

3. Pick a test point above or below the line and substitute the point into the inequality.

Choose (0, 0) because this point is easy to substitute into the inequality.

y > x + 3

(0) > (0) + 3

0 > 3    This is false!

Since the test point makes the inequality false, all points on that side of the line make the inequality false. Therefore we were correct to shade above the line. 10

2.3.1: Solving Linear Inequalities in Two Variables

Guided Practice - Example 2 Graph the solutions to the following inequality.

11

2.3.1: Solving Linear Inequalities in Two Variables

Guided Practice: Example 2, continued

1. Graph the inequality as a linear equation. Since the inequality is inclusive, use a solid line.

To graph the line, plot the y-intercept first, (0, 2). Then use the slope to find a second point. The slope is 3.

Count up three units and to the right one unit and plot a second point.

Connect the two points and extend the line to the edges of the coordinate plane.

12

2.3.1: Solving Linear Inequalities in Two Variables

2.3.1: Solving Linear Inequalities in Two Variables

1313

2.3.1: Solving Linear Inequalities in Two Variables

Guided Practice: Example 2, continued

14

2.3.1: Solving Linear Inequalities in Two Variables

2. Shade the appropriate area.Since the symbol is used we will shade above the y-intercept.

Guided Practice: Example 2, continued

3. Pick a test point above or below the line and substitute the point into the inequality.

Choose (0, 0) because this point is easy to substitute into the inequality.

(0) > 3(0) + 2

0 > 0 + 2

0 > 2    This is false!

Since the test point makes the inequality false, all points on that side of the line make the inequality false. Therefore we were correct to shade above the line

15

2.3.1: Solving Linear Inequalities in Two Variables

Guided Practice - Example 3 Graph the solutions to the following inequality.

16

2.3.1: Solving Linear Inequalities in Two Variables

Guided Practice: Example 3, continued

1. Graph the inequality as a linear equation. Since the inequality is not inclusive, use a dashed line.

To graph the line, plot the y-intercept first, (0, -1). Then use the slope to find a second point.

The slope is.

Count down two units and to the right three units and plot a second point.

Connect the two points and extend the line to the edges of the coordinate plane.

17

2.3.1: Solving Linear Inequalities in Two Variables

2.3.1: Solving Linear Inequalities in Two Variables

1818

2.3.1: Solving Linear Inequalities in Two Variables

Guided Practice: Example 3, continued

191919

2.3.1: Solving Linear Inequalities in Two Variables

2. Shade the appropriate area.Since the symbol is used we will shade below the y-intercept.

Guided Practice: Example 3, continued

3. Pick a test point above or below the line and substitute the point into the inequality.

Choose (0, 0) because this point is easy to substitute into the inequality.

(0) (0) - 1

0 0 - 1

0 -1    This is false!

Since the test point makes the inequality false, all points on that side of the line make the inequality false. Therefore we were correct to shade below the line

20

2.3.1: Solving Linear Inequalities in Two Variables

You Try!

Graph the following inequalities:

1. 2.

21