Section 5.6: Quadratic and rational inequalities
#1 – 8: Use the graph of f(x) to solve
a) f(x) > 0
b) f(x) < 0
1)
1)
First find the x-coordinates of all x-intercepts,
and in the case of fractions the x-coordinate of
the vertical asymptote.
x = -4, x = 3
Second plot on a number line with ±∞ and
create intervals with all round parenthesis as
neither question has an “or equal to”. The
intervals that correspond to where the graph is
above the x-axis go in part a. The regions that
correspond to where the graph is below the xaxis go in part b.
Above
Below
Above
(−∞, −4)
(-4, 3)
(3, ∞)
_____________________________
−∞
-4
3
∞
Answer
1a) f(x) > 0 (−∞, −𝟒) ∪ (𝟑, ∞)
1b) f(x) < 0 (-4,3)
3)
3)
First find the x-coordinates of all x-intercepts,
and in the case of fractions the x-coordinate of
the vertical asymptote.
x = -4, x = 3
Second plot on a number line with ±∞ and
create intervals with all round parenthesis as
neither question has an “or equal to”. The
intervals that correspond to where the graph is
above the x-axis go in part a. The regions that
correspond to where the graph is below the xaxis go in part b.
Below
Above
Below
(−∞, −4)
(-4, 3)
(3, ∞)
_____________________________
−∞
-4
3
∞
Answer
3a) f(x) < 0 (-4,3)
3b) f(x) > 0 (−∞, −𝟒) ∪ (𝟑, ∞)
Section 5.6: Quadratic and rational inequalities
5)
5)
First find the x-coordinates of all x-intercepts,
and in the case of fractions the x-coordinate of
the vertical asymptote.
x = -4, x = 3, x = 6
Second plot on a number line with ±∞ and
create intervals with all round parenthesis as
neither question has an “or equal to”. The
intervals that correspond to where the graph is
above the x-axis go in part a. The regions that
correspond to where the graph is below the xaxis go in part b.
Above
Below
Above Below
(−∞, −4)
(-4, 3)
(3,6)
(6, ∞)
_____________________________
−∞
-4
3
6
∞
Answer
5a) f(x) > 0 (−∞, −𝟒) ∪ (𝟑, 𝟔)
5b) f(x) < 0 (−𝟒, 𝟑) ∪ 𝟔, ∞)
7)
7)
First find the x-coordinates of all x-intercepts,
and in the case of fractions the x-coordinate of
the vertical asymptote.
x-coordinate of x-intercepts x = -2
x-coordinate of vertical asymptote x = 3
Second plot on a number line with ±∞ and
create intervals with all round parenthesis as
neither question has an “or equal to”. The
intervals that correspond to where the graph is
above the x-axis go in part a. The regions that
correspond to where the graph is below the xaxis go in part b.
Above
Below Above
(−∞, −2)
(-2, 3)
(3, ∞)
_____________________________
−∞
-2
3
∞
Answer
7a) f(x) > 0 (−∞, −𝟐) ∪ (𝟑, ∞)
7b) f(x) < 0 (-2,3)
Section 5.6: Quadratic and rational inequalities
#9 – 20: Use the graph of f(x) to solve
a) f(x) ≥ 0
b) f(x) ≤ 0
9)
9)
First find the x-coordinates of all x-intercepts,
and in the case of fractions the x-coordinate of
the vertical asymptote.
x-coordinate of x-intercepts x = -5,3,7
Second plot on number line with ±∞ and create
intervals. The x-intercepts get square brackets
because of the “or equal to.”
Below
Above
Below Above
(−∞, −5]
[-5, 3]
[3,7]
[7, ∞)
_____________________________
−∞
-5
3
7
∞
Answer:
9a) f(x) ≥ 0 [−𝟓, 𝟑] ∪ [𝟕, ∞)
9b) f(x) ≤ 0 (−∞, −𝟓] ∪ [𝟑, 𝟕]
11)
11)
First find the x-coordinates of all x-intercepts,
and in the case of fractions the x-coordinate of
the vertical asymptote.
x-coordinate of x-intercepts x = 2
no vertical asymptote.
Second plot on number line with ±∞ and create
intervals. The x-intercepts get square brackets
because of the “or equal to.”
Above
Above
(−∞, 2]
[2, ∞)
_____________________________
−∞
2
∞
Answer:
11a) f(x) ≥ 0 (−∞, 𝟐] ∪ [𝟐, ∞)
okay to write (−∞, ∞)
as the graph is always touching or above the xaxis
11b) f(x) ≤ 0 {2}
The graph is never below the x-axis, but it does
touch the x-axis when x=2 and this should be
noted in the answer to
part b.
Section 5.6: Quadratic and rational inequalities
13)
13)
First find the x-coordinates of all x-intercepts, and in the
case of fractions the x-coordinate of the vertical
asymptote.
x-coordinate of x-intercepts x = -5,3
no vertical asymptote.
Second plot on number line with ±∞ and create
intervals. The x-intercepts get square brackets because
of the “or equal to.”
Below
Above
Above
(−∞, −5]
[-5, 3]
[3, ∞)
_____________________________
−∞
-5
3
∞
Answer:
13a) f(x) ≥ 0 [-5, 3] ∪ [3, ∞)
13a) may also be written as [−5, ∞)
13b) f(x) ≤ 0 (−∞, −5] ∪ {3}
every x-intercept needs to be included in the answers to
part a and b. The single number x = 3 satisfies f(x) = 0 and
needs to part of the answer to 13b.
15)
15)
First find the x-coordinates of all x-intercepts, and in the
case of fractions the x-coordinate of the vertical
asymptote.
x-coordinate of x-intercepts x = 4
x-coordinate of vertical asymptote x = -5
Second plot on number line with ±∞ and create
intervals. The x-intercepts get square brackets because
of the “or equal to.”
The x-coordinate of every vertical asymptote always get
round brackets.
Above
Below
Above
(−∞, −5)
(-5, 4]
[4, ∞)
_____________________________
−∞
-5
4
∞
Answer:
15a) f(x) ≥ 0 (−∞, −𝟓) ∪ [𝟒, ∞)
15b) f(x) ≤ 0 (-5,4]
Section 5.6: Quadratic and rational inequalities
17)
17)
First find the x-coordinates of all x-intercepts,
and in the case of fractions the x-coordinate
of the vertical asymptote.
x-coordinate of x-intercepts x = 3
x-coordinate of vertical asymptote x = -6,-2
Second plot on number line with ±∞ and
create intervals. The x-intercepts get square
brackets because of the “or equal to.”
The x-coordinate of every vertical asymptote
always get round brackets.
Below
Above
Below Above
(−∞, −6)
(-6, -2)
(-2,3]
[3, ∞)
_____________________________
−∞
-6
-2
3
∞
Answer:
17a) f(x) ≥ 0 (−𝟔, −𝟐) ∪ [𝟑, ∞)
17b) f(x) ≤ 0 (−∞, −𝟔) ∪ (−𝟐, 𝟑]
19)
19)
First find the x-coordinates of all x-intercepts,
and in the case of fractions the x-coordinate of
the vertical asymptote.
No-x-intercepts
x-coordinate of vertical asymptote x = 0
Second plot on number line with ±∞ and
create intervals. The x-intercepts get square
brackets because of the “or equal to.”
The x-coordinate of every vertical asymptote
always get round brackets.
Below
Above
(−∞, 0)
(0, ∞)
_____________________________
−∞
0
∞
Answer:
19a) f(x) ≥ 0 (𝟎, ∞)
19b) f(x) ≤ 0 (−∞, 𝟎)
Section 5.6: Quadratic and rational inequalities
#21-32: Solve
a) f(x) > 0
b) f(x) < 0
21) 𝑓(𝑥) = 𝑥 2 + 5𝑥 − 6
21)
Sketch a graph using your
calculator.
First find the x-coordinates of all x-intercepts, and in the case of
fractions the x-coordinate of the vertical asymptote.
x2 + 5x - 6 = 0
(x + 6)(x – 1) = 0
x+6=0
x–1=0
x = -6
x=1
Second plot on a number line with ±∞ and create intervals
with all round parenthesis as neither question has an “or equal
to”. The intervals that correspond to where the graph is above
the x-axis go in part a. The regions that correspond to where
the graph is below the x-axis go in part b.
Above
Below
Above
(−∞, −6)
(-6, 1)
(1, ∞)
_____________________________
−∞
-6
1
∞
Answer
21a) f(x) > 0 (−∞, −𝟔)
21b) f(x) < 0 (-6, 1)
∪ (𝟏, ∞)
23) 𝑓(𝑥) = 9 − 𝑥 2
23)
Sketch a graph using your calculator.
First find the x-coordinates of all x-intercepts, and in the
case of fractions the x-coordinate of the vertical asymptote.
9 – x2 = 0
(3 + x)(3 – x) = 0
3+x=0
3–x =0
x = -3
x=3
Second plot on a number line with ±∞ and create intervals
with all round parenthesis as neither question has an “or
equal to”. The intervals that correspond to where the
graph is above the x-axis go in part a. The regions that
correspond to where the graph is below the x-axis go in
part b.
Below
Above
Below
(−∞, −3)
(-3, 3)
(3, ∞)
_____________________________
−∞
-3
3
∞
Answer
23a) f(x) > 0 (-3,3)
23b) f(x) < 0 (−∞, −𝟑)
∪ (𝟑, ∞)
25) 𝑓(𝑥) = 𝑥 3 + 6𝑥 2 − 9𝑥 − 54
25)
Sketch a graph using your calculator.
First find the x-coordinates of all x-intercepts, and
in the case of fractions the x-coordinate of the
vertical asymptote.
These x-intercepts require synthetic division. You
may find them be examining your graph to save a
time. I found the x-value of each x-intercept by
examining the graph.
x = -6, x = -3, x = 3
Second plot on a number line with ±∞ and
create intervals with all round parenthesis as
neither question has an “or equal to”. The
intervals that correspond to where the graph is
above the x-axis go in part a. The regions that
correspond to where the graph is below the xaxis go in part b.
Below
Above Below
Above
(−∞, −6) (-6,-3) (-3, 3)
(3, ∞)
___________________________________
−∞
-6
-3
3
∞
Answer
25a) f(x) > 0 (-6,3) ∪ (𝟑, ∞)
25b) f(x) < 0 (−∞, −𝟔)
∪ (−𝟑, 𝟑)
27) 𝑓(𝑥) = 𝑥 3 + 5𝑥 2 − 6𝑥
27)
Sketch a graph using your calculator.
First find the x-coordinates of all x-intercepts, and
in the case of fractions the x-coordinate of the
vertical asymptote.
x3 + 5x2 – 6x = 0
x(x2 + 5x – 6) = 0
x(x+6)(x-1) = 0
x=0
x=0
x+6=0
x = -6
x–1=0
x=1
Second plot on a number line with ±∞ and
create intervals with all round parenthesis as
neither question has an “or equal to”. The
intervals that correspond to where the graph is
above the x-axis go in part a. The regions that
correspond to where the graph is below the xaxis go in part b.
Below
Above Below
Above
(−∞, −6) (-6,0) (0, 1)
(1, ∞)
___________________________________
−∞
-6
0
1
∞
27a) f(x) > 0 (-6,0) ∪ (𝟏, ∞)
27b) f(x) < 0 (−∞, −𝟔)
∪ (𝟎, 𝟏)
29) 𝑓(𝑥) =
𝑥+3
𝑥−4
Sketch a graph using your calculator.
29)
First find the x-coordinates of all x-intercepts, and in the
case of fractions the x-coordinate of the vertical
asymptote
x-coordinate of x-intercept can be found by setting the
numerator equal to zero.
x+3=0
x = -3 (x-coordinate of x-intercept)
x-coordinate of vertical asymptote can be found by
setting the denominator equal to zero.
x–4= 0
x = 4 (x-coordinate of vertical asymptote)
Second plot on a number line with ±∞ and create
intervals with all round parenthesis as neither question
has an “or equal to”. The intervals that correspond to
where the graph is above the x-axis go in part a. The
regions that correspond to where the graph is below
the x-axis go in part b.
Above
Below
Above
(−∞, −3)
(-3, 4)
(4, ∞)
_____________________________
−∞
-3
4
∞
Answer
29a) f(x) > 0 (−∞, −𝟑)
29b) f(x) < 0 (-3, 4)
∪ (𝟒, ∞)
31) 𝑓(𝑥) =
𝑥−5
𝑥 2 −4
Sketch a graph using your calculator.
31)
First find the x-coordinates of all x-intercepts, and in
the case of fractions the x-coordinate of the vertical
asymptote
x-coordinate of x-intercept can be found by setting the
numerator equal to zero.
x-5=0
x = 5 (x-coordinate of x-intercept)
x-coordinate of vertical asymptote can be found by
setting the denominator equal to zero.
x2 – 4 = 0
(x + 2)(x – 2) = 0
x = -2, x = 2 (x-coordinates of vertical asymptote)
Second plot on a number line with ±∞ and create
intervals with all round parenthesis as neither
question has an “or equal to”. The intervals that
correspond to where the graph is above the x-axis go
in part a. The regions that correspond to where the
graph is below the x-axis go in part b.
Below
Above Below
Above
(−∞, −2) (-2,2) (2, 5)
(5, ∞)
___________________________________
−∞
-2
2
5
∞
Answer
31a) f(x) > 0 (-2,2) ∪ (𝟓, ∞)
31b) f(x) < 0 (−∞, −𝟐) ∪ (𝟐, 𝟓)
#33 – 42: Solve
a) f(x) ≥ 0
b) f(x) ≤ 0
33) 𝑓(𝑥) = 𝑥 2 − 5𝑥
33)
Sketch a graph using your calculator.
First find the x-coordinates of all x-intercepts, and in
the case of fractions the x-coordinate of the vertical
asymptote
x-coordinate of x-intercept
x2 – 5x = 0
x(x -5) = 0
x=0 x=5
Second plot on a number line with ±∞ and create
intervals with square brackets around the xintercepts and when there are vertical asymptotes,
round for the vertical asymptotes. The intervals that
correspond to where the graph is above the x-axis go
in part a. The regions that correspond to where the
graph is below the x-axis go in part b.
Above
Below
Above
(−∞, 0]
[0, 5]
[5, ∞)
_____________________________
−∞
0
5
∞
Answer:
33a) f(x) ≥ 0 (−∞, 0] ∪ [5, ∞)
33b) f(x) ≤ 0 [0,5]
35) 𝑓(𝑥) = (𝑥 − 2)2
35)
Sketch a graph using your calculator.
First find the x-coordinates of all x-intercepts, and in the case
of fractions the x-coordinate of the vertical asymptote
x-coordinate of x-intercept
x2 – 5x = 0
x(x -5) = 0
x=0 x=5
Second plot on a number line with ±∞ and create intervals
with square brackets around the x-intercepts and when there
are vertical asymptotes, round for the vertical asymptotes.
The intervals that correspond to where the graph is above the
x-axis go in part a. The regions that correspond to where the
graph is below the x-axis go in part b.
Above
Above
(−∞, 2]
[2, ∞)
_____________________________
−∞
2
∞
Answer:
35a) f(x) ≥ 0 (−∞, 𝟐] ∪ [𝟐, ∞) okay to write (−∞, ∞)
as the graph is always touching or above the x-axis
35b) f(x) ≤ 0 {2}
The graph is never below the x-axis, but it does touch the xaxis when x=2 and this should be noted in the answer to
part b.
37) 𝑓(𝑥) = (𝑥 + 5)(𝑥 − 3)2
37)
Sketch a graph using your calculator.
First find the x-coordinates of all x-intercepts, and in the
case of fractions the x-coordinate of the vertical
asymptote
x-coordinate of x-intercept
(x+5)(x-3)(x-3) = 0
x = -5 and x = 3
Second plot on a number line with ±∞ and create
intervals with square brackets around the x-intercepts
and when there are vertical asymptotes, round for the
vertical asymptotes. The intervals that correspond to
where the graph is above the x-axis go in part a. The
regions that correspond to where the graph is below the
x-axis go in part b.
Below
Above
Above
(−∞, −5]
[-5, 3]
[3, ∞)
_____________________________
−∞
-5
3
∞
Answer:
37a) f(x) ≥ 0 [−𝟓, 𝟑] ∪ [𝟑, ∞) which can be reduced to
[−𝟓, ∞)
37b) f(x) ≤ 0 need to include the point x = 3 along with
the interval as the graph touches the x-axis at x=3.
(−∞, −𝟓] ∪ {𝟑}
39) 𝑓(𝑥) =
𝑥+2
𝑥−6
Sketch a graph using your calculator.
39)
First find the x-coordinates of all x-intercepts, and in the
case of fractions the x-coordinate of the vertical
asymptote
x-coordinate of x-intercept can be found be setting the
numerator equal to zero.
x+2=0
x = -2 (x-coordinate of x-intercept)
x-coordinate of vertical asymptote can be found by
setting the denominator equal to zero.
x–6=0
x = 6 (x-coordinate of vertical asymptote)
Second plot on a number line with ±∞ and create
intervals with square brackets around the x-intercepts
and when there are vertical asymptotes, round for the
vertical asymptotes. The intervals that correspond to
where the graph is above the x-axis go in part a. The
regions that correspond to where the graph is below the
x-axis go in part b.
Above
Below
Above
(−∞, −2]
[-2, 6)
(6, ∞)
_____________________________
−∞
-2
6
∞
Answer:
39a) f(x) ≥ 0 (−∞, −2] ∪ (6, ∞)
39b) f(x) ≤ 0 [-2,6)
41) 𝑓(𝑥) =
𝑥+6
𝑥 2 −9
Sketch a graph using your calculator.
41)
First find the x-coordinates of all x-intercepts, and in the
case of fractions the x-coordinate of the vertical
asymptote
x-coordinate of x-intercept can be found be setting the
numerator equal to zero.
x+6=0
x = -6 (x-coordinate of x-intercept)
x-coordinate of vertical asymptote can be found by
setting the denominator equal to zero.
x2 – 9 = 0
(x+3)(x-3) = 0
x = -3, x = 3 (x-coordinate of vertical asymptote)
Second plot on a number line with ±∞ and create
intervals with square brackets around the x-intercepts
and when there are vertical asymptotes, round for the
vertical asymptotes. The intervals that correspond to
where the graph is above the x-axis go in part a. The
regions that correspond to where the graph is below
the x-axis go in part b.
Below
Above Below
Above
(−∞, −6] [-6,-3) (-3, 3)
(3, ∞)
___________________________________
−∞
-6
-3
3
∞
Answer:
41a) f(x) ≥ 0 [−6 − 3) ∪ (3, ∞)
41b) f(x) ≤ 0 (−∞, −6] ∪ (−3,3)