Graphs of Linear Equations
CHAPTER
11
11.1
11.2
11.3
11.4
11.5
Graphs and Applications of Linear Equations
More with Graphing and Intercepts
Slope and Applications
Equations of Lines
Graphing Using the Slope
and the y-Intercept
11.6 Parallel and Perpendicular Lines
11.7 Graphing Inequalities in Two Variables
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 2
11.7
Graphing Inequalities in Two
Variables
OBJECTIVES
a Determine whether an ordered pair of numbers is
a solution of an inequality in two variables.
b Graph linear inequalities.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 3
11.7
Graphing Inequalities in Two
Variables
Determine whether an ordered pair of numbers is
a
a solution of an inequality in two variables.
A graph of an inequality is a drawing that represents its
solutions. An inequality in one variable can be graphed on
the number line. An inequality in two variables can be
graphed on a coordinate plane.
The solutions of inequalities in two variables are ordered
pairs.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 4
11.7
Graphing Inequalities in Two
Variables
Determine whether an ordered pair of numbers is
a
a solution of an inequality in two variables.
EXAMPLE 1
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 5
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 3
We first graph the line y = x. Every solution of y = x is an
ordered pair like (3, 3) in which the first and second
coordinates are the same. We draw the line y = x dashed
because its points are not solutions of y > x.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 6
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 3
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 7
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 3
Several ordered pairs are plotted in the half-plane above
the line y = x. Each is a solution of y > x. We can check a
pair such as (–2, 4) as follows:
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 8
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 3
It turns out that any point on the same side of y = x as (–2,
4) is also a solution. If we know that one point in a halfplane is a solution, then all points in that half-plane are
solutions. We could have chosen other points to check.
The graph of y > x is shown on the next slide. (Solutions
are indicated by color shading throughout.) We shade the
half-plane above y = x.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 9
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 3
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 10
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
A linear inequality is one that we can get from a linear
equation by changing the equals symbol to an inequality
symbol. Every linear equation has a graph that is a straight
line. The graph of a linear inequality is a halfplane,
sometimes including the line along the edge.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 11
11.7
Graphing Inequalities in Two
Variables
To graph an inequality in two variables:
1. Replace the inequality symbol with an equals sign and
graph this related linear equation.
2. If the inequality symbol is < or > draw the line dashed.
If the inequality symbol is or , draw the line solid.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 12
11.7
Graphing Inequalities in Two
Variables
3. The graph consists of a half-plane, either above or
below or left or right of the line, and, if the line is solid,
the line as well. To determine which half-plane to shade,
choose a point not on the line as a test point. Substitute
to find whether that point is a solution of the inequality.
If it is, shade the half-plane containing that point. If it is
not, shade the half-plane on the opposite side of the
line.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 13
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 5
1. First, we graph the line 2x + 3y = 6. The intercepts are
(0, 2) and (3, 0).
2. Since the inequality contains the symbol, we draw the
line solid to indicate that any pair on the line is a solution.
3. Next, we choose a test point that is not on the line. We
substitute to determine whether this point is a solution.
The origin is generally an easy point to use:
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 14
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 5
We see that (0, 0) is a solution, so we shade the lower
half-plane. Had the substitution given us a false
inequality, we would have shaded the other half-plane.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 15
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 6
There is no y-term in this inequality, but we can rewrite this
inequality as x + 0y < 3. We use the same technique that
we have used with the other examples.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 16
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 6
1. We graph the related equation x = 3 on the plane.
2. Since the inequality symbol is < we use a dashed line.
3. The graph is a half-plane either to the left or to the
right of the line x = 3. To determine which, we consider a
test point, (–4, 5):
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 17
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 6
We see that (–4, 5) is a
solution, so all the pairs in the
half-plane containing (–4, 5)
are solutions. We shade that
half-plane.
The solutions are all those
ordered pairs whose first
coordinates are less than 3.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 18
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 7
1. We first graph y = –4.
2. We use a solid line to
indicate that all points on the
line are solutions.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 19
11.7
Graphing Inequalities in Two
Variables
b Graph linear inequalities.
EXAMPLE 7
3. We then use (2, 3) as a test point and substitute:
Since (2, 3) is a solution, all points in the half-plane
containing (2, 3) are solutions. Note that this half-plane
consists of all ordered pairs whose second coordinate is
greater than or equal to –4.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 20