1001 Algebra Problems.PDF
- 20. This can be verified by choosing an
arbitrary point in the shaded region, say (0,0),
and observing that we can verify this by substi-
tuting a point from the shaded region, such as
(0,0) into the inequality, resulting in the true
statement –20.
- a.The graph of the line is solid, so we know
that it is included in the solution set, and that
the inequality describing the shaded region
must include equality (either or ). Next,
since the graph of the line rises from left to
right at the rate of three vertical unit up per
one horizontal unit right, its slope is 3. It
crosses the y-axis at (0,1), so the equation of
the line is y= 3x+ 1. Finally, since the shaded
region is above the liney= 3x+ 1, the inequal-
ity illustrated by this graph isy 3 x+ 1. This
can be verified by choosing any point in the
shaded region, such as (2,0), and substituting
it into the inequality, which results in the true
statement 2 1.
- d.The fact that the graph of the line is solid
means that it is included in the solution set, so
the inequality describing the shaded region
must include equality (that is, it must be either
or ). Next, since the graph of the line falls
from left to right at the rate of two vertical
unit down per one horizontal unit right, its
slope is –2. And, since it crosses the y-axis at
(0,4), we conclude that the equation of the line
is y= –2x+ 4. Finally, since the shaded region
is above the liney = –2x+ 4, we conclude that
the inequality illustrated by this graph isy
–2x+ 4. Observe that simplifying 3x– y
7 x+
y– 8 results in this inequality. This can be veri-
fied by choosing a point in the shaded region,
such as (0,5), substituting it into the inequality
to produce the true statement 5 4.
- d.Substituting x= 3 and y = –2 into the
inequality 9x– 1yyields the true statement
26 –2. We can therefore conclude that
(3,–2) satisfies this inequality.
- d.First, since the given inequality does not
include equality, the horizontal line y= 4 is
not included in the solution set and should be
dashed. Because y4, any point in the solu-
tion set (the shaded region) must have a y-
coordinate that is larger than 4. Such points
occur only above the line y= 4. The correct
graph is given by choice d.
- c.Since the given inequality does not include
equality, the vertical line x= 4 is not included
in the solution set and should be dashed. Also,
since x4, any point in the solution set (the
shaded region) must have an x-coordinate that
is larger than 4. Such points occur to the right
of the line x= 4. The correct graph is shown in
choice c.
- b.The graph of the line is solid, so it is included
in the solution set, and the inequality describing
the shaded region must include equality (it must
be either or≤). Next, since the graph of the
line falls from left to right at the rate of one ver-
tical unit down per seven horizontal units right,
its slope is –^17 . It crosses the y-axis at (0,10), so
the equation of the line is y= –^17 x+ 10. Finally,
since the shaded region is below the line y= –^17
x+ 10, the inequality illustrated by this graph is
y – ^17 x+ 10. Observe that simplifying
–28y 2 x– 14(y + 10) results in this inequal-
ity. This can be verified by choosing any point
in the shaded region, such as (0,5), and substi-
tuting it into the inequality to produce the true
statement 5 10.
- b.The following graph illustrates the inequal-
ity y ( 2x+ 7, whose solution set intersects
all four quadrants.
ANSWERS & EXPLANATIONS–
