Transcript Linear Inequalities

Graphing a Linear Inequality
Graphing a linear inequality is very
similar to graphing a linear
equation.
Graphing a Linear Inequality
• 1) Solve the inequality for y
(or for x if there is no y).
• 2) Change the inequality to an equation
and graph.
• 3) If the inequality is < or >, the line
is dotted. If the inequality is ≤ or
≥, the line is solid.
Linear Inequalities
• A linear inequality in two variables can
be written in any one of these forms:




Ax + By < C
Ax + By > C
Ax + By ≤ C
Ax + By ≥ C
• An ordered pair (x, y) is a solution of the
linear inequality if the inequality is TRUE
when x and y are substituted into the
inequality.
Example 1
•
Which ordered pair is a solution of
5x - 2y ≤ 6?
A.
B.
C.
D.
(0, -3)
(5, 5)
(1, -2)
(3, 3)
Graphing Linear
Inequalities
• The graph of a linear inequality is
the set of all points in a coordinate
plane that represent solutions of
the inequality.
– We represent the boundary line of
the inequality by drawing the
function represented in the
inequality.
Graphing Linear
Inequalities
The boundary line will be a:
– Solid line when ≤ and ≥ are used.
– Dashed line when < and > are used.
Our graph will be shaded on one
side of the boundary line to show
where the solutions of the
inequality are located.
Graphing Linear
Inequalities
Here are some steps to help graph linear
inequalities:
1. Graph the boundary line for the inequality.
Remember:


≤ and ≥ will use a solid curve.
< and > will use a dashed curve.
2. Test a point NOT on the boundary line to determine
which side of the line includes the solutions. (The
origin is always an easy point to test, but make sure
your line does not pass through the origin)


If your test point is a solution (makes a TRUE statement),
shade THAT side of the boundary line.
If your test points is NOT a solution (makes a FALSE
statement), shade the opposite side of the boundary line.
Example 2
• Graph the inequality x ≤ 4 in a coordinate
plane.
• HINT: Remember HOY VEX.
• Decide whether to
use a solid or
dashed line.
• Use (0, 0) as a
test point.
• Shade where the
solutions will be.
y
5
x
-5
-5
5
Graphing a Linear Inequality
• Graph the inequality 3 - x > 0
• First, solve the inequality for x.
3-x>0
-x > -3
x<3
Graph: x<3
• Graph the line x = 3.
• Because x < 3 and
not x ≤ 3, the line
will be dotted.
• Now shade the side
of the line where
x < 3 (to the left of
the line).
6
4
2
3
Graphing a Linear Inequality
• 4) To check that the shading is correct, pick a
point in the area and plug it into the
inequality.
• 5) If the inequality statement is true, the
shading is correct. If the inequality
statement is false, the shading is incorrect.
Graphing a Linear Inequality
• Pick a point, (1,2),
in the shaded area.
• Substitute into the
original inequality
3–x>0
3–1>0
2>0
• True! The inequality
has been graphed
correctly.
6
4
2
3
Example 3
• Graph 3x - 4y > 12 in a coordinate plane.
• Sketch the boundary line of the graph.
 Find the x- and
y-intercepts and
plot them.
• Solid or dashed
line?
• Use (0, 0) as a
test point.
• Shade where the
solutions are.
y
5
x
-5
-5
5
Example 4:
Using a new Test Point
• Graph y < 2/5x in a coordinate plane.
• Sketch the boundary line of the graph.
 Find the x- and y-intercept and plot them.
5
 Both are the origin!
y
• Use the line’s slope
to graph another point.
• Solid or dashed
line?
• Use a test point
OTHER than the
origin.
• Shade where the
solutions are.
-5
x
-5
5
Absolute Value Functions and
Graphs
Graphing Absolute Value Functions
Absolute Value Functions
• An absolute value function is a function with an
absolute value as part of the equation…
f(x) = |mx + b|
• Graphs of absolute value equations have two special
properties:
a) a vertex
b) they look like angles
Absolute Value Functions
Vertex – point where
the graph changes
direction
Finding the Vertex
For an equation y = |mx + b| + c,
vertex
=
-b , c
m
Example: Find the vertex of y = |4x + 2| - 3
Finding the Vertex
For an equation y = |mx + b| + c,
vertex
Example:
Answer:
=
-b , c
m
Find the vertex of y = |4x + 2| - 3
= -2 , -3
4
= -1 , -3
2
Absolute Value Functions
Steps to graphing an absolute value function…
1. Find the vertex
2. Write two linear equations and find slope
3. Use slope to plot points, connect the dots
Absolute Value Functions
Example 1:
Graph y = |3x + 12|.
Absolute Value Functions
Step 1: Find the vertex
y
=
|3x + 12|
m = -b =
c=
Absolute Value Functions
Step 1: Find the vertex
y
=
|3x + 12|
m=3
-b = -12
c=0
Absolute Value Functions
Step 1: Find the vertex
y
=
|3x + 12|
m=3
vertex
=
-b , c
m
vertex
=
-12 , 0
3
vertex =
(-4, 0)
-b = -12
c=0
Absolute Value Functions
Step 1: Find the vertex
vertex = (-4, 0)
Absolute Value Functions
Step 2:
Write two linear equations and
find slope.
y = |3x + 12|
Positive
Negative
Absolute Value Functions
Step 2:
Write two linear equations and
find slope.
y = |3x + 12|
Positive
y = 3x + 12
m1 =
y
Negative
= -3x – 12
m2 =
Absolute Value Functions
Step 3: Use the slope to plot points
vertex = (-4, 0)
m1 = 3
m2 = -3
Absolute Value Functions
Step 3: Use the slope to plot points
vertex = (-4, 0)
m1 = 3
m2 = -3
Absolute Value Functions
Step 3: Use the slope to plot points
vertex = (-4, 0)
m1 = 3
m2 = -3
Absolute Value Functions
Step 3: Use the slope to plot points
vertex = (-4, 0)
m1 = 3
m2 = -3
Absolute Value Functions
Step 3: Use the slope to plot points
vertex = (-4, 0)
m1 = 3
m2 = -3
Absolute Value Functions
Example 2:
Graph y = |3x + 6| - 2
Absolute Value Functions
Step 1:
Find the vertex
y = |3x + 6| - 2
Absolute Value Functions
Step 1:
Find the vertex
y
= |3x + 6| - 2
m = 3 -b = -6 c = -2
vertex
= (-6/3, -2)
= (-2, -2)
Absolute Value Functions
Step 1:
Find the vertex
vertex = (-2, -2)
Absolute Value Functions
Step 2:
Write two linear equations and
find slope
y = |3x + 6| - 2
Positive
Negative
Absolute Value Functions
Step 2:
Write two linear equations and
find slope
y = |3x + 6| - 2
Positive
y + 2 = 3x + 6
y
= 3x + 4
Negative
y + 2 = -3x – 6
y
= -3x – 8
Absolute Value Functions
Step 2:
Write two linear equations and
find slope
y = |3x + 6| - 2
Positive
y + 2 = 3x + 6
y
= 3x + 4
m
= 3
Negative
y + 2 = -3x – 6
y
= -3x – 8
m
= -3
Absolute Value Functions
Step 3: Use the slope to plot points
vertex = (-2, -2)
m1 = -3
m2 = 3
Absolute Value Functions
Step 3: Use the slope to plot points
vertex = (-2, -2)
m1 = -3
m2 = 3
Absolute Value Functions
Step 3: Use the slope to plot points
vertex = (-2, -2)
m1 = -3
m2 = 3
Absolute Value Functions
Step 3: Use the slope to plot points
vertex = (-2, -2)
m1 = -3
m2 = 3
Absolute Value Functions
Step 3: Use the slope to plot points
vertex = (-2, -2)
m1 = -3
m2 = 3
Absolute Value Functions
Step 3: Use the slope to plot points
vertex = (-2, -2)
m1 = -3
m2 = 3