Good Morning Systems of Inequalities. Holt McDougal Algebra 1 Solving Linear Inequalities Warm Up Graph each inequality. 1. x > –5 2. y ≤ 0 3. Write –6x

Good MorningSystems of Inequalities

Holt McDougal Algebra 1

Solving Linear Inequalities

Warm UpGraph each inequality.1. x > –5 2. y ≤ 0

3. Write –6x + 2y = –4 in slope-intercept form, and graph. y = 3x – 2



Graph and solve linear inequalities in two variables.

Objective



linear inequalitysolution of a linear inequality

Vocabulary



A linear inequality is similar to a linear equation, but the equal sign is replaced with an inequality symbol. A solution of a linear inequality is any ordered pair that makes the inequality true.



Tell whether the ordered pair is a solution of the inequality.

Example 1A: Identifying Solutions of Inequalities

(–2, 4); y < 2x + 1

Substitute (–2, 4) for (x, y).

y < 2x + 1

4 2(–2) + 1

4 –4 + 14 –3<

(–2, 4) is not a solution.



A linear inequality describes a region of a coordinate plane called a half-plane. All points in the region are solutions of the linear inequality. The boundary line of the region is the graph of the related equation.





Graphing Linear Inequalities

Step 1 Solve the inequality for y (slope-intercept form).

Step 2Graph the boundary line. Use a solid line for ≤ or ≥. Use a dashed line for < or >.

Step 3Shade the half-plane above the line for y > or ≥. Shade the half-plane below the line for y < or y ≤. Check your answer.



Graph the solutions of the linear inequality.

Example 2A: Graphing Linear Inequalities in Two Variables

y 2x – 3

Step 1 The inequality is already solved for y.

Step 2 Graph the boundary line y = 2x – 3. Use a solid line for .

Step 3 The inequality is , so shade below the line.



The point (0, 0) is a good test point to use if it does not lie on the boundary line.

Helpful Hint




Example 2B: Graphing Linear Inequalities in Two Variables

5x + 2y > –8

Step 1 Solve the inequality for y.

5x + 2y > –8 –5x –5x

2y > –5x – 8

y > x – 4

Step 2 Graph the boundary line Use a dashed line for >.

y = x – 4.



Step 3 The inequality is >, so shade above the line.

Example 2B Continued

Graph the solutions of the linear inequality.5x + 2y > –8




Substitute ( 0, 0) for (x, y) because it is not on the boundary line.

The point (0, 0) satisfies the inequality, so the graph is correctly shaded.

Check

y > x – 4

0 (0) – 4

0 –40 –4>

Graph the solutions of the linear inequality.5x + 2y > –8




Example 2C: Graphing Linear Inequalities in two Variables

4x – y + 2 ≤ 0


4x – y + 2 ≤ 0

–y ≤ –4x – 2

–1 –1

y ≥ 4x + 2

Step 2 Graph the boundary line y ≥= 4x + 2. Use a solid line for ≥.



Step 3 The inequality is ≥, so shade above the line.

Example 2C Continued


4x – y + 2 ≤ 0



Check It Out! Example 2a


4x – 3y > 12


4x – 3y > 12 –4x –4x

–3y > –4x + 12

y < – 4

Step 2 Graph the boundary line y = – 4.

Use a dashed line for <.



Check It Out! Example 2a Continued

Step 3 The inequality is <, so shade below the line.


4x – 3y > 12



Check It Out! Example 2a Continued

Substitute ( 1, –6) for (x, y) because it is not on the boundary line.

The point (1, –6) satisfies the inequality, so the graph is correctly shaded.

Check

y < – 4

–6 (1) – 4 –6 – 4

–6 <


4x – 3y > 12



Check It Out! Example 2b


2x – y – 4 > 0


2x – y – 4 > 0

– y > –2x + 4

y < 2x – 4

Step 2 Graph the boundary line

y = 2x – 4. Use a dashed line for <.



Check It Out! Example 2b Continued

Step 3 The inequality is <, so shade below the line.


2x – y – 4 > 0





2x – y – 4 > 0

Substitute (3, –3) for (x, y) because it is not on the boundary line.

The point (3, –3) satisfies the inequality, so the graph is correctly shaded.

Check

–3 2(3) – 4

–3 6 – 4

–3 < 2

y < 2x – 4



Check It Out! Example 2c


Step 1 The inequality is already solved for y.

Step 3 The inequality is ≥, so shade above the line.

Step 2 Graph the boundary

line . Use a solid line for

≥.

=



Check It Out! Example 2c Continued

Check

y ≥ x + 1

0 (0) + 1

0 0 + 1

0 ≥ 1

A false statement means that the half-plane containing

(0, 0) should NOT be shaded. (0, 0) is not one of the solutions, so the graph is shaded correctly.

Graph the solutions of the linear inequality.Substitute (0, 0) for (x, y) because it

is not on the boundary line.



Write an inequality to represent the graph.

Example 3A: Writing an Inequality from a Graph

y-intercept: 1; slope:

Write an equation in slope-intercept form.

The graph is shaded above a dashed boundary line.

Replace = with > to write the inequality



Write an inequality to represent the graph.

Example 3B: Writing an Inequality from a Graph

y-intercept: –5 slope:

Write an equation in slope-intercept form.

The graph is shaded below a solid boundary line.

Replace = with ≤ to write the inequality



Example 2A: Solving a System of Linear Inequalities by Graphing

Graph the system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions.

y ≤ 3

y > –x + 5

y ≤ 3 y > –x + 5

Graph the system.

(8, 1) and (6, 3) are solutions.

(–1, 4) and (2, 6) are not solutions.

(6, 3)

(8, 1)

(–1, 4)

(2, 6)



Example 2B: Solving a System of Linear Inequalities by Graphing


–3x + 2y ≥ 2

y < 4x + 3

–3x + 2y ≥ 2 Solve the first inequality for y.

2y ≥ 3x + 2



y < 4x + 3

Graph the system.



(0, 0) and (–4, 5) are not solutions.

(2, 6)

(1, 3)

(0, 0)

(–4, 5)





y ≤ x + 1 y > 2

y ≤ x + 1 y > 2

Graph the system.


(–3, 1) and (–1, –4) are not solutions.

(3, 3)

(4, 4)

(–3, 1)

(–1, –4)





y > x – 7 3x + 6y ≤ 12

Solve the second inequality for y.

3x + 6y ≤ 12

6y ≤ –3x + 12

y ≤ x + 2




Graph the system.

y > x − 7

y ≤ – x + 2

(0, 0) and (3, –2) are solutions.

(4, 4) and (1, –6) are not solutions.

(4, 4)

(1, –6)

(0, 0)

(3, –2)



In Lesson 6-4, you saw that in systems of linear equations, if the lines are parallel, there are no solutions. With systems of linear inequalities, that is not always true.



Graph the system of linear inequalities.Describe the solutions.

Example 3A: Graphing Systems with Parallel Boundary Lines

y ≤ –2x – 4 y > –2x + 5

This system has no solutions.



Graph the system of linear inequalities. Describe the solutions.

Example 3B: Graphing Systems with Parallel Boundary Lines

y > 3x – 2

y < 3x + 6

The solutions are all points between the parallel lines but not on the dashed lines.




Example 3C: Graphing Systems with Parallel Boundary Lines

y ≥ 4x + 6 y ≥ 4x – 5

The solutions are the same as the solutions of y ≥ 4x + 6.




y > x + 1 y ≤ x – 3


This system has no solutions.




y ≥ 4x – 2 y ≤ 4x + 2


The solutions are all points between the parallel lines including the solid lines.




y > –2x + 3 y > –2x

Check It Out! Example 3c

The solutions are the same as the solutions of y > –2x + 3.

Documents

Good Morning Systems of Inequalities. Holt McDougal Algebra 1 Solving Linear Inequalities Warm Up Graph each inequality. 1. x > –5 2. y ≤ 0 3. Write –6x