Download linear inequality

Good Morning
Systems of Inequalities
Solving Linear Inequalities
Warm Up
Graph each inequality.
1. x > –5
3. Write –6x + 2y = –4
in slope-intercept form,
and graph.
y = 3x – 2
Holt McDougal Algebra 1
2. y ≤ 0
Solving Linear Inequalities
Objective
Graph and solve linear inequalities in
two variables.
Holt McDougal Algebra 1
Solving Linear Inequalities
Vocabulary
linear inequality
solution of a linear inequality
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
Example 1A: Identifying Solutions of Inequalities
Tell whether the ordered pair is a solution of
the inequality.
(–2, 4); y < 2x + 1
y < 2x + 1
4 2(–2) + 1
4 –4 + 1
4 < –3 
(–2, 4) is not a solution.
Holt McDougal Algebra 1
Substitute (–2, 4) for (x, y).
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 (slopeintercept form).
Step 2
Graph the boundary line. Use a solid line
for ≤ or ≥. Use a dashed line for < or >.
Shade the half-plane above the line for y >
Step 3 or ≥. Shade the half-plane below the line
for y < or y ≤. Check your answer.
Holt McDougal Algebra 1
Solving Linear Inequalities
Example 2A: Graphing Linear Inequalities in Two
Variables
Graph the solutions of the linear inequality.
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
Helpful Hint
The point (0, 0) is a good test point to use if it
does not lie on the boundary line.
Holt McDougal Algebra 1
Solving Linear Inequalities
Example 2B: Graphing Linear Inequalities in Two
Variables
Graph the solutions of the linear inequality.
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 y =
dashed line for >.
Holt McDougal Algebra 1
x – 4. Use a
Solving Linear Inequalities
Example 2B Continued
Graph the solutions of the linear inequality.
5x + 2y > –8
Step 3 The inequality is >, so
shade above the line.
Holt McDougal Algebra 1
Solving Linear Inequalities
Example 2B Continued
Graph the solutions of the linear inequality.
5x + 2y > –8
Substitute ( 0, 0)
Check y >
x–4
for (x, y)
because it is
not on the
0
(0) – 4
boundary line.
0
–4
The point (0, 0)
0 > –4
satisfies the
inequality, so the
graph is correctly
shaded.
Holt McDougal Algebra 1
Solving Linear Inequalities
Example 2C: Graphing Linear Inequalities in two
Variables
Graph the solutions of the linear inequality.
4x – y + 2 ≤ 0
Step 1 Solve the inequality for y.
4x – y + 2 ≤ 0
–y
–1
≤ –4x – 2
–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
Example 2C Continued
Graph the solutions of the linear inequality.
4x – y + 2 ≤ 0
Step 3 The inequality is ≥, so
shade above the line.
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 =
Use a dashed line for <.
Holt McDougal Algebra 1
– 4.
Solving Linear Inequalities
Check It Out! Example 2a Continued
Graph the solutions of the linear inequality.
4x – 3y > 12
Step 3 The inequality is <, so
shade below the line.
Holt McDougal Algebra 1
Solving Linear Inequalities
Check It Out! Example 2a Continued
Graph the solutions of the linear inequality.
4x – 3y > 12
Check
y<
–6
–6
–6 <
–4
(1) – 4
–4

Substitute ( 1, –6) for (x, y)
because it is not on the
boundary line.
Holt McDougal Algebra 1
The point (1, –6) satisfies the
inequality, so the graph is
correctly shaded.
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
Graph the solutions of the linear inequality.
2x – y – 4 > 0
Step 3 The inequality is <, so
shade below the line.
Holt McDougal Algebra 1
Solving Linear Inequalities
Check It Out! Example 2b Continued
Graph the solutions of the linear inequality.
2x – y – 4 > 0
Check
y < 2x – 4
–3
2(3) – 4
–3
6–4
–3 < 2

Substitute (3, –3) for (x, y)
because it is not on the
boundary line.
Holt McDougal Algebra 1
The point (3, –3) satisfies the
inequality, so the graph is
correctly shaded.
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 2 Graph the boundary
line
=
. Use a solid line for
≥.
Step 3 The inequality is ≥,
so shade above the line.
Holt McDougal Algebra 1
Solving Linear Inequalities
Check It Out! Example 2c Continued
Graph the solutions of the linear inequality.
Substitute (0, 0) for (x, y) because it
is not on the boundary line.
Check
y≥
0
x+1
(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.
Holt McDougal Algebra 1
Solving Linear Inequalities
Example 3A: Writing an Inequality from a Graph
Write an inequality to represent the graph.
y-intercept: 1; slope:
Write an equation in slopeintercept form.
The graph is shaded above a
dashed boundary line.
Replace = with > to write the inequality
Holt McDougal Algebra 1
Solving Linear Inequalities
Example 3B: Writing an Inequality from a Graph
Write an inequality to represent the graph.
y-intercept: –5 slope:
Write an equation in slopeintercept 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

(2, 6)
(–1, 4)
y > –x + 5


(6, 3)
Graph the system.
y≤3
y > –x + 5
(8, 1) and (6, 3) are solutions.
(–1, 4) and (2, 6) are not solutions.
Holt McDougal Algebra 1
(8, 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
2y ≥ 3x + 2
Holt McDougal Algebra 1
Solve the first inequality for y.
Solving Linear Inequalities
Example 2B Continued
Graph the system.
y < 4x + 3


(–4, 5)

(2, 6) and (1, 3) are solutions.
(0, 0) and (–4, 5) are not solutions.
Holt McDougal Algebra 1
(2, 6)
(0, 0)

(1, 3)
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.
(4, 4)
y≤x+1
y>2

(3, 3)
(–3, 1)
Graph the system.
y≤x+1
y>2

(–1, –4)
(3, 3) and (4, 4) are solutions.
(–3, 1) and (–1, –4) are not solutions.
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
3x + 6y ≤ 12
Solve the second inequality
6y ≤ –3x + 12 for y.
y≤
Holt McDougal Algebra 1
x+2
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)


(0, 0)

(3, –2)

(1, –6)
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
Example 3A: Graphing Systems with Parallel
Boundary Lines
Graph the system of linear inequalities.
Describe the solutions.
y ≤ –2x – 4
y > –2x + 5
This system has
no solutions.
Holt McDougal Algebra 1
Solving Linear Inequalities
Example 3B: Graphing Systems with Parallel
Boundary Lines
Graph the system of linear inequalities.
Describe the solutions.
y < 3x + 6
y > 3x – 2
The solutions are all points
between the parallel lines but
not on the dashed lines.
Holt McDougal Algebra 1
Solving Linear Inequalities
Example 3C: Graphing Systems with Parallel
Boundary Lines
Graph the system of linear inequalities.
Describe the solutions.
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
Check It Out! Example 3a
Graph the system of linear inequalities.
Describe the solutions.
y>x+1
y≤x–3
This system has
no solutions.
Holt McDougal Algebra 1
Solving Linear Inequalities
Check It Out! Example 3b
Graph the system of linear inequalities.
Describe the solutions.
y ≥ 4x – 2
y ≤ 4x + 2
The solutions are all
points between the
parallel lines including
the solid lines.
Holt McDougal Algebra 1
Solving Linear Inequalities
Check It Out! Example 3c
Graph the system of linear inequalities.
Describe the solutions.
y > –2x + 3
y > –2x
The solutions are the
same as the solutions of
y > –2x + 3.
Holt McDougal Algebra 1