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UNIT 4 TOPICS
Solving application problems dealing
with variation
Graphing linear equations
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 1
Variation
Variation is an equation that relates one
variable to one or more other variables.
 k is the constant of proportionality
 If a variable y varies directly with a variable x,
then y = kx
 If a variable y varies inversely with
a variable, x, then
k

y=

x
The general form of a joint variation, where y,
varies directly as x and z, is y = kxz
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Slide 6 - 2
Example
Suppose y varies inversely as x. If y = 12 when x = 18,
find y when x = 21.
k

y
x
k
12 
18
216  k

Now substitute 216 for k, and find y when x = 21.
k
y
x
216
y
21
y  10.3
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Slide 6 - 3
Symbols of Inequality




Recall < , ≤, >, ≥ are inequality symbols.
3x – 8 < 10 and x > 3 are examples of
inequalities.
Same rules for solving as for equations except
you change the direction of the inequality
symbol when multiplying or dividing both sides
of an inequality by a negative number.
We can state the solution by solving or show it
by graphing.
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Slide 6 - 4
Example: Graphing


Graph the solution set of x ≤ 4, where x is a real
number, on the number line.
The numbers less than or equal to 4 are all the
points on the number line to the left of 4 and 4
itself. The closed circle at 4 shows that 4 is
included in the solution set.
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Slide 6 - 5
Example: Graphing


Graph the solution set of x > 3, where x is a real
number, on the number line.
The numbers greater than 3 are all the points
on the number line to the right of 3. The open
circle at 3 is used to indicate that 3 is not
included in the solution set.
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Slide 6 - 6
Example: Solve and graph the solution. You try.

Solve 3x – 8 < 10 and graph the solution set.
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Slide 6 - 7
Example: Solve and graph the solution

Solve 3x – 8 < 10 and graph the solution set.
3 x  8  10
3 x  8  8  10  8
3 x  18
3 x 18

3
3
x6

The solution set is all real numbers less than 6.
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 8
Compound Inequality

Graph the solution set of the inequality
-4 < x ≤ 3
b) where x is a real number

The solution set consists of all real numbers
between -4 and 3, including the 3 but not the -4.
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Slide 6 - 9
Solve –2x + 6 ≥ 4x – 10. You try. Can you put in
set builder notation from Unit 2?
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Slide 6 - 10
Graph the solution set of –2x + 6 ≥ 4x – 10 on the
real number line.
b.
a.
8
3
d.
c.
8
3
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8
8
Slide 6 - 11
Graph the solution set of –2x + 6 ≥ 4x – 10 on the
real number line.
b.
a.
8
3
d.
c.
8
3
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8
8
Slide 6 - 12
Rectangular Coordinate System



y-axis
The horizontal line is
called the x-axis.
Quadrant I
The vertical line is Quadrant II
called the y-axis.
The point of
x-axis
intersection is the
origin
origin.
Quadrant III Quadrant IV
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Slide 6 - 13
Plotting Points


Each point in the
xy-plane corresponds
to a unique ordered
pair (a, b).
Plot the point B = (2,
4).Move 2 units right
Move 4 units up
4 units
2 units
Lets Plot C=(-4,-2);
D=(-3,4); E=(1,-5)
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Slide 6 - 14
Graphing Linear Equations – one way is to plot 2
points. (page 163)

Graph the equation
y = 5x + 2
x
0
2/5
1
y
2
0
3
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Slide 6 - 15