Good Morning Systems of Inequalities. Holt McDougal Algebra 1 Solving Linear Inequalities Warm Up...

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

Holt McDougal Algebra 1

Solving Linear Inequalities

Graph and solve linear inequalities in two variables.

Objective

Holt McDougal Algebra 1

Solving Linear Inequalities

linear inequalitysolution of a linear inequality

Vocabulary

Holt McDougal Algebra 1

Solving Linear Inequalities

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.

Holt McDougal Algebra 1

Solving Linear Inequalities

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.

Holt McDougal Algebra 1

Solving Linear Inequalities

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.

Holt McDougal Algebra 1

Solving Linear Inequalities

Holt McDougal Algebra 1

Solving Linear Inequalities

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.

Holt McDougal Algebra 1

Solving Linear Inequalities

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.

Holt McDougal Algebra 1

Solving Linear Inequalities

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

Helpful Hint

Holt McDougal Algebra 1

Solving Linear Inequalities

Graph the solutions of the linear inequality.

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.

Holt McDougal Algebra 1

Solving Linear Inequalities

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

Example 2B Continued

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

Holt McDougal Algebra 1

Solving Linear Inequalities

Example 2B Continued

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

Holt McDougal Algebra 1

Solving Linear Inequalities

Graph the solutions of the linear inequality.

Example 2C: Graphing Linear Inequalities in two Variables

4x – y + 2 ≤ 0

Step 1 Solve the inequality for y.

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 ≥.

Holt McDougal Algebra 1

Solving Linear Inequalities

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

Example 2C Continued

Graph the solutions of the linear inequality.

4x – y + 2 ≤ 0

Holt McDougal Algebra 1

Solving Linear Inequalities

Check It Out! Example 2a

Graph the solutions of the linear inequality.

4x – 3y > 12

Step 1 Solve the inequality for y.

4x – 3y > 12 –4x –4x

–3y > –4x + 12

y < – 4

Step 2 Graph the boundary line y = – 4.

Use a dashed line for <.

Holt McDougal Algebra 1

Solving Linear Inequalities

Check It Out! Example 2a Continued

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

Graph the solutions of the linear inequality.

4x – 3y > 12

Holt McDougal Algebra 1

Solving Linear Inequalities

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 <

Graph the solutions of the linear inequality.

4x – 3y > 12

Holt McDougal Algebra 1

Solving Linear Inequalities

Check It Out! Example 2b

Graph the solutions of the linear inequality.

2x – y – 4 > 0

Step 1 Solve the inequality for y.

2x – y – 4 > 0

– y > –2x + 4

y < 2x – 4

Step 2 Graph the boundary line

y = 2x – 4. Use a dashed line for <.

Holt McDougal Algebra 1

Solving Linear Inequalities

Check It Out! Example 2b Continued

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

Graph the solutions of the linear inequality.

2x – y – 4 > 0

Holt McDougal Algebra 1

Solving Linear Inequalities

Check It Out! Example 2b Continued

Graph the solutions of the linear inequality.

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

Holt McDougal Algebra 1

Solving Linear Inequalities

Check It Out! Example 2c

Graph the solutions of the linear inequality.

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

≥.

=

Holt McDougal Algebra 1

Solving Linear Inequalities

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.

Holt McDougal Algebra 1

Solving Linear Inequalities

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

Holt McDougal Algebra 1

Solving Linear Inequalities

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

Holt McDougal Algebra 1

Solving Linear Inequalities

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)

Holt McDougal Algebra 1

Solving Linear Inequalities

Example 2B: 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.

–3x + 2y ≥ 2

y < 4x + 3

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

2y ≥ 3x + 2

Holt McDougal Algebra 1

Solving Linear Inequalities

y < 4x + 3

Graph the system.

Example 2B Continued

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

(0, 0) and (–4, 5) are not solutions.

(2, 6)

(1, 3)

(0, 0)

(–4, 5)

Holt McDougal Algebra 1

Solving Linear Inequalities

Check It Out! Example 2a

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

y ≤ x + 1 y > 2

y ≤ x + 1 y > 2

Graph the system.

(3, 3) and (4, 4) are solutions.

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

(3, 3)

(4, 4)

(–3, 1)

(–1, –4)

Holt McDougal Algebra 1

Solving Linear Inequalities

Check It Out! Example 2b

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

y > x – 7 3x + 6y ≤ 12

Solve the second inequality for y.

3x + 6y ≤ 12

6y ≤ –3x + 12

y ≤ x + 2

Holt McDougal Algebra 1

Solving Linear Inequalities

Check It Out! Example 2b Continued

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)

Holt McDougal Algebra 1

Solving Linear Inequalities

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.

Holt McDougal Algebra 1

Solving Linear Inequalities

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.

Holt McDougal Algebra 1

Solving Linear Inequalities

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.

Holt McDougal Algebra 1

Solving Linear Inequalities

Graph the system of linear inequalities. Describe the solutions.

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.

Holt McDougal Algebra 1

Solving Linear Inequalities

Graph the system of linear inequalities. Describe the solutions.

y > x + 1 y ≤ x – 3

Check It Out! Example 3a

This system has no solutions.

Holt McDougal Algebra 1

Solving Linear Inequalities

Graph the system of linear inequalities. Describe the solutions.

y ≥ 4x – 2 y ≤ 4x + 2

Check It Out! Example 3b

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

Holt McDougal Algebra 1

Solving Linear Inequalities

Graph the system of linear inequalities. Describe the solutions.

y > –2x + 3 y > –2x

Check It Out! Example 3c

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