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1
Grade 10
/ product inequalities and sign tables
55 exercises (100% corrected)
a
Graphical reading of sign tables
(2 exercices)
Sign table of affine functions
(3 exercices)
Construction of sign tables
(3 exercices)
Product Specifications Table
(5 exercices)
Table of Signs for a Quotient
(3 exercices)
Inequations and sign tables
(9 exercices)
Graphical solutions to inequalities and sign chart
(3 exercices)
Graphical readings and algebraic manipulations
(2 exercices)
Relative positions of curves: graphic resolution
(3 exercices)
Relative positions of curves: algebraic solution
(2 exercices)
Problems
(5 exercices)
Deepening: reflection on algebraic operations
(2 exercices)
Further study: systems of inequalities
(3 exercices)
Deepening: manipulation of frames
(3 exercices)
image PNG : /home/_math/_exercice/d0/471/metapost/exo_faux.png
-4
-2
0
2
4
-4
-2
2
4
C
f
-6
-4
-2
0
2
-2
2
4
6
C
g
-6
-4
-2
0
2
4
6
-4
-2
2
4
C
f
-6
-4
-2
0
2
4
6
-4
-2
2
4
C
g
x
O
y
x
0
(
d
Co
e`cien
t
directeur
p
ositif
(
m>
0
)
x
O
y
x
0
(
d
Co
e`cien
t
directeur
négatif
(
m<
0
)
x
Signe
de
f
(
x
−∞
∞
x
0
0
−
x
Signe
de
f
(
x
−∞
∞
x
0
0
−
ChingQuizz
:
9
exercises
a
v
ailable
for
Quizz
assessmen
t
:
1.
Gr
aphic
al
r
e
ading
of
sign
tables
E.472
Sho
wn
b
elo
w
are
the
graphical
rep-
resen
tations
of
the
functions
f
and
g
defined
resp
ectiv
ely
on
[
−
4
;
4]
and
[
−
6
;
2]
.
1
Determine,
graphically
,
the
solutions
of
the
inequations
:
f
(
x
)
0
;
g
(
x
)
0
2
Dra
w
up
the
sign
tables
for
the
functions
f
and
g
:
x
−
4
4
f
(
x
)
x
−
6
2
g
(
x
)
E.481
Consider
the
t
w
o
functions
f
and
g
defined
on
−
6
;
6
whose
graphical
represen
tations
are
giv
en
b
elo
w
:
Dra
w
up
the
sign
tables
of
the
functions
f
and
g
on
−
6
;
6
2.
Sign
table
of
affine
functions
E.9784
Prop
osal
:
1
Consider
the
linear
function
f
defined
b
y
:
f
(
x
)
=
2
x
+
3
Determine
the
sign
table
of
the
function
f
.
2
Consider
the
linear
function
f
defined
b
y
:
g
(
x
)
=
−
x
+
1
Determine
the
sign
table
of
the
function
g
.
E.9785
1
a
Consider
the
function
f
affine
defined
b
y
:
f
(
x
)
=
3
x
−
4
Complete
the
sign
table
for
the
function
f
:
x
−∞
+
∞
f
(
x
)
b
Consider
the
function
g
affine
defined
b
y
:
g
(
x
)
=
−
2
x
+
1
Complete
the
sign
table
for
the
function
g
:
x
−∞
+
∞
g
(
x
)
2
Let
h
b
e
the
function
obtained
b
y
the
pro
duct
of
the
functions
f
and
g
.
That
is,
for
an
y
x
∈
R
:
h
(
x
)
=
f
(
x
)
×
g
(
x
)
Sp
ecify
whether
the
follo
wing
statemen
ts
are
true
or
false
:
a
h
(
−
5)
is
p
ositif
b
h
(
−
1)
is
p
ositif
c
h
(1)
is
p
ositif
d
h
(3.2)
is
p
ositif
E.9786
Consider
an
affine
fonctin
f
defined
on
R
and
whose
sign
table
is
giv
en
b
elo
w
:
x
−∞
1
2
+
∞
f
(
x
)
+
0
−
F
urthermore,
the
inequation
f
(
x
)
3
admits
for
solutions
the
set
S
:
S
=
−∞
;
−
3
Determine
the
slop
e-in
tercept
formof
the
function
f
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/472
sacados/472
-4
-2
0
2
4
-4
-2
2
4
C
f
-6
-4
-2
0
2
-2
2
4
6
C
g
chapExoCorrec/481
sacados/481
-6
-4
-2
0
2
4
6
-4
-2
2
4
C
f
-6
-4
-2
0
2
4
6
-4
-2
2
4
C
g
chapExoCorrec/9784
sacados/9784
x
O
y
x
0
(
d
Co
e`cien
t
directeur
p
ositif
(
m>
0
)
x
O
y
x
0
(
d
Co
e`cien
t
directeur
négatif
(
m<
0
)
x
Signe
de
f
(
x
−∞
∞
x
0
0
−
x
Signe
de
f
(
x
−∞
∞
x
0
0
−
chapExoCorrec/9785
sacados/9785
chapExoCorrec/9786
sacados/9786
-
1
2
2
-
∞
∞
-
1
2
2
-
∞
∞
-
1
2
2
-
∞
∞
2
x
1
x
−
2
(2
x
x
−
-
∞
∞
-
∞
∞
-
∞
∞
3
x
−
4
3
−
x
g
(
x
3.
Construction
of
sign
tables
E.468
1
Consider
the
function
f
defined
b
y
the
relation
:
f
(
x
)
=
2
x
2
−
3
x
−
2
In
this
question,
w
e
will
study
the
sign
of
the
function
f
.
a
Establish
equalit
y
:
f
(
x
)
=
(2
x
+1)(
x
−
2)
.
b
Solv
e
the
follo
wing
t
w
o
inequations
:
2
x
+
1
<
0
;
x
−
2
<
0
c
In
the
table
b
elo
w
and
for
the
t
w
o
factors
2
x
+1
and
x
−
2
,
color
:
in
blue
the
in
terv
als
o
v
er
whic
h
the
factor
is
p
ositiv
e
;
in
red
the
in
terv
als
o
v
er
whic
h
the
factor
is
negativ
e.
d
Complete
the
third
line
using
the
sign
rule
for
a
pro
d-
uct.
e
Solv
e
the
inequation
:
f
(
x
)
0
.
2
Consider
the
function
g
whose
image
of
a
n
um
b
er
x
is
giv
en
b
y
the
relation
:
g
(
x
)
=
−
3
x
2
+
13
x
−
12
a
Establish
the
follo
wing
equalit
y
:
g
(
x
)
=
(3
x
−
4)(3
−
x
)
b
As
in
the
previous
question,
complete
the
table
b
elo
w
:
c
Deduce
the
set
of
solutions
to
the
inequation
g
(
x
)
<
0
E.469
1
a
Solv
e
at
R
the
follo
wing
inequations
:
4
−
2
x
0
;
5
x
+
15
0
b
Deduce
the
solutions
of
the
inequations
:
4
−
2
x
<
0
;
5
x
+
15
<
0
2
a
In
the
first
t
w
o
lines
of
the
sign
table,
indicate
the
sign
of
the
expressions
on
R
:
x
−∞
+
∞
4
−
2
x
5
x
+15
(4
−
2
x
)(5
x
+15)
b
Complete
the
third
ro
w
of
the
table
to
indicate
the
sign
of
the
pro
duct
(4
−
2
x
)(5
x
+15)
on
R
.
c
Deduce
the
set
of
solutions
to
the
inequation
:
(4
−
2
x
)(5
x
+
15)
0
E.4380
Consider
the
functions
f
and
g
whose
images
of
the
n
um
b
er
x
are
resp
ectiv
ely
defined
b
y
:
f
(
x
)
=
2
−
x
×
x
−
5
;
g
(
x
)
=
(2
−
x
)(
x
−
5)
1
a
Justify
that
the
function
f
is
not
defined
for
the
n
um
b
er
3
.
b
Determine
the
definition
set
of
the
function
f
.
2
a
Determine
the
image
of
the
n
um
b
er
3
b
y
the
func-
tion
g
.
b
Determine
the
definition
set
of
the
function
g
.
4.
Pr
o
duct
Sp
e
cific
ations
T
able
E.4455
Complete
the
sign
tables
b
elo
w
:
1
x
−∞
+
∞
2
x
+
1
3
+
x
(2
x
+1)(3+
x
)
2
x
−∞
+
∞
2
−
x
4
x
−
3
(2
−
x
)(4
x
−
3)
E.4876
Complete
the
sign
tables
b
elo
w
:
1
x
−∞
+
∞
1
−
x
2
x
+
1
(1
−
x
)(2
x
+1)
2
x
−∞
+
∞
x
−
3
−
2
x
+
4
(
x
−
3)(
−
2
x
+4)
h
ttps:/
/c
hingmath.fr
chapExoCorrec/468
sacados/468
-
1
2
2
-
∞
∞
-
1
2
2
-
∞
∞
-
1
2
2
-
∞
∞
2
x
1
x
−
2
(2
x
x
−
-
∞
∞
-
∞
∞
-
∞
∞
3
x
−
4
3
−
x
g
(
x
chapExoCorrec/469
sacados/469
chapExoCorrec/4380
sacados/4380
chapExoCorrec/4455
sacados/4455
chapExoCorrec/4876
sacados/4876
E.460
Establish
the
sign
table
for
algebraic
expressions
:
a
(
x
+
1)(2
−
x
)
b
−
(2
x
+
4)(
x
−
2)
c
(
x
+
1)
2
E.11648
Construire
le
tableau
de
signes
de
c
ha-
cune
des
expressions
ci-dessous
:
a
(2
x
−
15)(2
x
−
3)
b
x
(
x
+
1)
E.11650
Construire
le
tableau
de
signe
des
ex-
pressions
:
a
(
−
2
x
−
1)(
x
+
1)
b
(
−
x
+
2)(
x
+
3)
5.
T
able
of
Signs
for
a
Quotient
E.9787
Complete
the
sign
tables
b
elo
w
:
1
x
−∞
+
∞
x
+
5
−
2
x
−
8
x
+
5
−
2
x
−
8
2
x
−∞
+
∞
x
−
1
4
−
x
−
x
−
1
(
x
−
1)(4
−
x
)
−
x
−
1
E.9788
Complete
the
sign
tables
b
elo
w
:
1
x
−∞
+
∞
2
+
x
2
−
x
2
+
x
2
−
x
2
x
−∞
+
∞
4
x
+
1
x
−
1
x
(4
x
+1)(
x
−
1)
x
E.11712
Construire
le
tableau
de
signe
des
ex-
pressions
:
a
5
x
−
3
4
x
+
7
b
−
2
x
+
5
6
x
+
15
6.
Ine
quations
and
sign
tables
E.480
Solv
e
the
follo
wing
inequalities
:
a
(
x
+
4)(1
−
2
x
)
0
b
3
+
x
2
−
x
0
E.9819
Solv
e
the
follo
wing
inequalities
:
a
(3
−
2
x
)(5
x
+
2)
0
b
(3
x
+
1)(4
−
2
x
)
0
E.9790
Solv
e
the
inequation
:
−
(
x
+
1)(
x
−
2)
1
−
x
>
0
E.9820
Solv
e
inequalities
:
a
(5
−
2
x
)(
x
−
1)
3
x
−
2
>
0
b
−
(2
x
+
1)
2
(4
x
−
3)(1
−
2
x
)
0
E.465
1
Expand
:
(2
x
+
1)(
x
−
1)
2
Solv
e
the
follo
wing
inequation
:
2
x
2
−
x
−
1
x
2
+
1
0
.
E.442
1
Expand
:
(
x
−
1)(
x
−
5)
2
Solv
e
:
(
x
−
3)
2
−
4
3
−
2
x
<
0
Hin
t
:
w
e
will
use
the
result
of
question
1
E.445
Consider
the
function
f
defined
b
y
the
follo
wing
relationship
:
f
:
x
↦−
→
x
+
1
−
2
x
2
+
7
x
−
3
1
Establish
the
follo
wing
equalit
y
:
−
2
x
2
+
7
x
−
3
=
(2
x
−
1)(3
−
x
)
2
Giv
e
the
definition
set
of
the
function
f
.
3
a
Dra
w
up
the
sign
table
for
the
function
f
on
its
def-
inition
set.
b
Solv
e
the
inequation
:
f
(
x
)
0
E.11713
Résoudre
les
inéquations
suiv
an
tes
:
a
(
x
+
4)(5
−
2
x
)
0
b
(3
−
x
)(3
−
6
x
)
<
0
E.11714
Résoudre
les
inéquations
suiv
an
tes
:
a
5
x
+
2
x
−
3
0
b
5
−
2
x
x
+
4
0
h
ttps:/
/c
hingmath.fr
chapExoCorrec/460
sacados/460
chapExoCorrec/11648
sacados/11648
chapExoCorrec/11650
sacados/11650
chapExoCorrec/9787
sacados/9787
chapExoCorrec/9788
sacados/9788
chapExoCorrec/11712
sacados/11712
chapExoCorrec/480
sacados/480
chapExoCorrec/9819
sacados/9819
chapExoCorrec/9790
sacados/9790
chapExoCorrec/9820
sacados/9820
chapExoCorrec/465
sacados/465
chapExoCorrec/442
sacados/442
chapExoCorrec/445
sacados/445
chapExoCorrec/11713
sacados/11713
chapExoCorrec/11714
sacados/11714
-2
-1
2
3
4
I
-1
2
J
O
C
f
-6
-5
-4
-3
-2
-1
2
3
4
5
6
7
8
I
-1
2
3
J
O
C
g
x
-6
-5
-4
-3
-2
-1
0
1
2
3
y
-2
-1
1
2
3
C
f
-3
-2
-1
0
1
2
3
-4
-2
2
4
C
f
7.
Gr
aphic
al
solutions
to
ine
qualities
and
sign
chart
E.5027
Consider
the
function
f
defined
on
the
in
terv
al
−
2
;
4
,
whose
represen
tativ
e
curv
e
C
f
is
giv
en
in
the
orthonormal
co
ordinate
system
O
;
I
;
J
b
elo
w
:
1
W
e
will
lea
v
e
the
construction
lines
necessary
for
solving
the
follo
wing
questions
:
a
Graphically
determine
the
image
of
the
n
um
b
er
2
b
y
the
function
f
.
b
Graphically
determine
the
an
teceden
ts
of
the
n
um
b
er
1
b
y
the
function
f
.
2
a
Graphically
determine
the
set
of
solutions
to
the
in-
equalit
y
f
(
x
)
0
b
Complete
the
sign
table
for
the
function
f
:
x
f
(
x
)
E.365
In
an
orthonormal
co
ordinate
system
(
O
;
I
;
J
)
,
consider
the
curv
e
C
g
represen
ting
the
function
g
:
1
Giv
e,
without
justification,
the
domain
of
the
function
g
.
2
a
Graphically
,
solv
e
the
equation
f
(
x
)
=
0
.
b
Complete
the
sign
table
for
the
function
f
.
x
f
(
x
)
E.9656
Consider
a
function
f
whose
rep-
resen
tativ
e
curv
e
C
f
is
giv
en
in
the
orthonormal
co
ordinate
system
b
elo
w
:
1
Giv
e
the
domain
of
the
function
f
.
2
a
Graphically
,
giv
e
the
solutions
to
the
equation
f
(
x
)
=
0
.
b
Dra
w
up
the
sign
table
for
the
function
f
.
8.
Gr
aphic
al
r
e
adings
and
algebr
aic
manipulations
E.458
Consider
the
function
f
defined
on
R
b
y
the
relation
:
f
(
x
)
=
x
3
−
3
x
+
2
1
Graphically
,
determine
the
set
of
solutions
to
the
inequa-
tion
f
(
x
)
>
0
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5027
sacados/5027
-2
-1
2
3
4
I
-1
2
J
O
C
f
chapExoCorrec/365
sacados/365
-6
-5
-4
-3
-2
-1
2
3
4
5
6
7
8
I
-1
2
3
J
O
C
g
chapExoCorrec/9656
sacados/9656
x
-6
-5
-4
-3
-2
-1
0
1
2
3
y
-2
-1
1
2
3
C
f
chapExoCorrec/458
sacados/458
-3
-2
-1
0
1
2
3
-4
-2
2
4
C
f
-2
-1
0
1
2
3
-4
-2
2
4
C
f
-7
-6
-5
-4
-3
-2
-1
2
3
4
5
6
7
I
-4
-3
-2
-1
2
3
4
J
O
C
f
C
g
-2
-1
2
3
4
I
-2
2
4
J
O
2
a
Establish
the
follo
wing
equalit
y
:
f
(
x
)
=
(
x
+2)(
x
−
1)
2
b
Dra
w
up
the
sign
table
for
the
function
f
at
R
.
c
Deduce
the
set
of
solutions
to
the
inequation
:
f
(
x
)
>
0
E.446
Consider
the
function
f
whose
image
of
a
n
um
b
er
x
is
defined
b
y
the
expression
:
f
(
x
)
=
−
x
3
+
2
x
2
+
x
−
2
In
the
plane
pro
vided
with
an
orthogonal
reference
frame,
w
e
ha
v
e
dra
wn
the
curv
e
C
f
represen
tativ
e
of
the
function
f
:
1
By
graphical
reading,
giv
e
the
in
terv
als
o
v
er
whic
h
the
function
f
is
strictly
p
ositiv
e.
2
Sho
w
the
equalit
y
:
−
x
3
+2
x
2
+
x
−
2
=
−
(
x
−
2)(
x
2
−
1)
3
a
Dra
w
up
the
table
of
signs
for
f
.
b
Deduce
the
solutions
of
the
inequation
:
f
(
x
)
>
0
.
9.
R
elative
p
ositions
of
curves
:
gr
aphic
r
esolution
E.462
In
the
plane
pro
vided
with
a
refer-
ence
frame
O
;
I
;
J
,
consider
the
t
w
o
curv
es
C
f
and
C
g
rep-
resen
tativ
e
resp
ectiv
ely
of
the
functions
f
and
g
defined
on
−
7
;
7
:
1
Determine,
graphically
,
the
co
ordinates
of
the
in
tersec-
tion
p
oin
ts
of
the
curv
es
C
f
and
C
g
.
2
Graphically
,
solv
e
the
inequation
:
g
(
x
)
f
(
x
)
E.372
Consider
the
function
f
whose
rep-
resen
tation
is
giv
en
b
elo
w
in
the
orthogonal
reference
frame
O
;
I
;
J
)
:
W
e
are
in
terested
in
the
affine
function
g
defined
b
y
the
rela-
tion
:
g
:
x
↦−
→
x
+
1
1
Dra
w
the
represen
tativ
e
curv
e
of
the
function
g
in
the
ab
o
v
e
reference
frame.
2
Graphically
,
solv
e
the
equation
:
f
(
x
)
=
g
(
x
)
.
3
Graphically
solv
e
the
inequation
:
f
(
x
)
g
(
x
)
h
ttps:/
/c
hingmath.fr
chapExoCorrec/446
sacados/446
-2
-1
0
1
2
3
-4
-2
2
4
C
f
chapExoCorrec/462
sacados/462
-7
-6
-5
-4
-3
-2
-1
2
3
4
5
6
7
I
-4
-3
-2
-1
2
3
4
J
O
C
f
C
g
chapExoCorrec/372
sacados/372
-2
-1
2
3
4
I
-2
2
4
J
O
-4
-3
-2
-1
2
3
4
I
-3
-2
-1
2
3
4
J
O
C
f
C
g
-2
-1
2
3
4
I
-6
-5
-4
-3
-2
-1
2
3
J
O
C
f
C
g
1
x
y
x
1
41
E.2142
In
the
plane
pro
vided
with
an
or-
thonormal
reference
frame,
w
e
giv
e
the
represen
tativ
e
curv
es
of
the
functions
f
and
g
defined
on
the
in
terv
al
[
−
4
;
4]
:
Consider
the
inequation
:
f
(
x
)
<
g
(
x
)
1
Whic
h
of
the
n
um
b
ers
b
elo
w
are
solutions
of
this
inequa-
tion
:
a
−
2.5
b
−
0.25
c
1
2
Solv
e
this
inequation
graphically
.
10.
R
elative
p
ositions
of
curves
:
algebr
aic
solution
E.4915
1
In
the
plane
pro
vided
with
a
reference
frame
O
;
I
;
J
orthogonal,
consider
the
represen
tativ
e
curv
es
C
f
and
C
g
represen
tativ
e
of
the
functions
f
and
g
.
Solv
e
grahpically
on
the
in
terv
al
−
2
;
4
the
follo
wing
inequation
:
f
(
x
)
<
g
(
x
)
2
The
functions
f
and
g
are
defined
on
R
b
y
the
expres-
sions
:
f
(
x
)
=
x
3
−
3
x
2
−
x
+
1
;
g
(
x
)
=
−
x
2
+
2
x
+
1
a
Determine
the
expression
of
the
p
olynomial
P
of
degree
1
v
erifying
the
equalit
y
:
x
3
−
2
x
2
−
3
x
=
(
x
2
−
3
x
)
×
P
b
Deduce
on
whic
h
in
terv
als
the
curv
e
C
g
is
strictly
ab
o
v
e
C
f
.
E.8192
In
the
plane
pro
vided
with
a
refer-
ence
frame
O
;
I
;
J
orthogonal,
consider
the
represen
tativ
e
curv
es
C
f
and
C
g
represen
tativ
e
of
the
functions
f
and
g
where
the
functions
f
and
g
are
defined
b
y
:
f
(
x
)
=
6
x
3
+
2
x
2
+
x
+
1
;
g
(
x
)
=
2
x
2
+
19
x
+
13
1
Determine
the
reals
a
and
b
realizing
the
iden
tit
y
:
6
x
3
−
18
x
−
12
=
(2
x
+2)(3
x
+3)(
ax
+
b
)
2
Deduce
on
whic
h
in
terv
als
the
curv
e
C
g
is
strictly
ab
o
v
e
C
f
.
11.
Pr
oblems
E.2868
A
man
ufacturer
of
cardb
oard
b
o
xes
uses
rolls
to
pro
duce
a
41
cm
wide
strip
of
cardb
oard
from
whic
h
he
traces
and
cuts
out
b
o
x
patterns
b
efore
gluing
them
on.
He
arranges
his
patterns
as
sho
wn
in
the
dra
wing
b
elo
w
:
h
ttps:/
/c
hingmath.fr
chapExoCorrec/2142
sacados/2142
-4
-3
-2
-1
2
3
4
I
-3
-2
-1
2
3
4
J
O
C
f
C
g
chapExoCorrec/4915
sacados/4915
-2
-1
2
3
4
I
-6
-5
-4
-3
-2
-1
2
3
J
O
C
f
C
g
chapExoCorrec/8192
sacados/8192
chapExoCorrec/2868
sacados/2868
1
x
y
x
1
41
x
9
cm
4
cm
A
B
C
D
E
F
5
m
10
m
x
A
B
C
D
E
F
G
The
b
o
xes,
in
the
shap
e
of
straigh
t
blo
c
ks,
ha
v
e
t
w
o
x
cm
-
square
faces,
fitted
with
t
w
o
1
cm
-wide
tabs
for
gluing,
and
four
other
faces
whose
dimensions
in
cm
are
x
and
y
,
as
w
ell
as
a
flap
for
closure.
1
a
Giv
e
an
expression
for
y
as
a
function
of
x
.
b
Justify
that
the
v
alue
of
x
b
elongs
to
the
in
terv
al
0
;
19.5
.
2
Demonstrate
that
the
v
olume
V
,
in
cm
3
,
of
the
b
o
x
is
giv
en,
as
a
function
of
x
,
b
y
the
form
ula
:
V
=
39
x
2
−
2
x
3
3
a
Determine
the
expression
of
the
p
olynomial
P
v
eri-
fying
the
equalit
y
:
39
x
2
−
2
x
3
−
972
=
(
x
−
18)(
x
−
6)
×
P
b
Deduce
the
v
alues
of
x
for
whic
h
the
v
olume
V
is
greater
than
972
cm
3
.
4
a
Determine
the
expression
of
the
p
olynomial
Q
v
eri-
fying
the
equalit
y
:
V
(
x
)
−
2197
=
(
−
2
x
−
13)
×
Q
b
Deduce
the
sign
table
for
the
expression
:
V
(
x
)
−
2197
.
c
Giv
e
the
maxim
um
v
olume
the
man
ufacturer
can
ob-
tain
with
this
t
yp
e
of
b
o
x
;
for
what
v
alue
of
x
is
this
maxim
um
reac
hed?
E.1979
Consider
the
triangle
AB
C
righ
t-
angled
A
suc
h
that
:
AB
=
9
cm
;
AC
=
4
cm
Consider
a
p
oin
t
D
b
elonging
to
segmen
t
[
AB
]
and
note
x
the
length
of
segmen
t
[
B
D
]
.
F
rom
the
p
oin
t
D
w
e
construct
a
rectangle
D
E
F
A
suc
h
that
:
E
∈
[
B
C
]
;
F
∈
[
AC
]
Note
A
the
area
of
the
rectangle
D
E
F
A
.
1
a
Determine
the
expression
for
length
F
A
as
a
func-
tion
of
x
.
b
Justify
,
briefly
,
that
the
real
n
um
b
er
x
b
elongs
to
the
in
terv
al
0
;
9
.
2
Establish
that
the
area
A
is
expressed
as
a
function
of
x
b
y
:
A
(
x
)
=
4
x
−
4
9
·
x
2
3
a
Dra
w
up
the
table
of
v
ariations
of
the
function
A
on
the
in
terv
al
0
;
9
.
b
What
is
the
maxim
um
area
reac
hed
b
y
the
A
area?
4
W
e
wish
to
kno
w
the
v
alues
of
x
for
whic
h
the
area
A
has
greater
than
or
equal
to
5
cm
2
:
a
Establish
the
follo
wing
factorization
:
−
4
x
2
+
36
x
−
45
=
(2
x
−
15)(3
−
2
x
)
b
Solv
e
the
inequation
:
A
(
x
)
5
.
E.11721
1
a
Etablir
la
factorisation
:
−
3
x
2
+10
x
−
3
=
(
x
−
3)(1
−
3
x
)
b
Résoudre
l’inéquation
:
−
3
x
2
+
10
x
−
3
>
0
2
On
considère
la
figure
ci-dessous
comp
osée
des
deux
rect-
angles
AB
C
D
et
AE
F
G
:
Déterminer
p
our
quelle
v
aleur
de
x
,
l’aire
de
la
partie
hac
h
urée
est
strictemen
t
sup
érieure
à
53
m
2
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/1979
sacados/1979
x
9
cm
4
cm
A
B
C
D
E
F
chapExoCorrec/11721
sacados/11721
5
m
10
m
x
A
B
C
D
E
F
G
3
m
9
m
x
A
B
C
D
E
F
G
A
B
C
D
E
F
G
H
I
2
x
x
4
cm
4
cm
1
1
x
2
1
1
x
·
(1
x
2
·
(1
x
1
2
2
x
−
1
2
x
−
1
2
x
E.11722
1
a
Etablir
la
factorisation
:
−
2
x
2
+9
x
−
10
=
(
x
−
2)(5
−
2
x
)
b
Résoudre
l’inéquation
:
−
2
x
2
+
9
x
−
10
0
2
On
considère
la
figure
ci-dessous
comp
osée
des
deux
rect-
angles
AB
C
D
et
AE
F
G
:
Déterminer
p
our
quelle
v
aleur
de
x
,
l’aire
de
la
partie
hac
h
urée
est
strictemen
t
sup
érieure
à
37
m
2
.
E.11723
1
a
Etablir
la
factorisation
:
4
x
2
−
12
x
+
27
4
=
4
x
−
9
x
−
3
4
b
Résoudre
l’inéquation
:
4
x
2
−
12
x
+
27
4
0
2
On
considère
la
figure
ci-dessous
comp
osée
:
du
carré
AE
F
G
,
de
deux
rectangles
AB
C
D
et
C
I
F
H
.
Les
p
oin
ts
B
,
D
,
I
,
H
ap-
partiennen
t
aux
côtés
du
carré
AE
F
G
.
On
considère
le
domaine
grisé
représen
té
ci-con
tre
et
on
note
son
aire
A
:
(les
mesur
es
sont
exprimé
es
en
c
entimètr
e)
Déterminer
l’ensem
ble
des
v
aleurs
de
x
réalisan
t
l’inéquation
:
A
37
4
Indication
:
toute
trace
de
rec
herc
he
ou
de
prise
d’initiativ
e
sera
prise
en
compte
dans
l’év
aluation.
12.
De
ep
ening
:
r
efle
ction
on
algebr
aic
op
er
ations
E.471
Here’s
a
studen
t’s
solution
to
an
in-
equation
:
1
is
−
2
the
solution
to
the
inequation
:
1
1+
x
2
?
a
b
What
can
w
e
deduce
from
the
prop
osed
resolution?
2
W
e
break
do
wn
the
studen
t’s
resolution,
indicating
the
algebraic
manipulation
p
erformed
at
eac
h
step
b
y
the
studen
t.
Complete
the
diagram
b
elo
w
:
3
a
What
can
b
e
the
sign
of
the
expression
1+
x
?
b
Deduce
the
error
made
b
y
the
studen
t.
4
Solv
e
the
inequation
:
1
1+
x
2
E.479
A
studen
t
has
pro
duced
the
follo
wing
solution
to
an
inequation
:
1
is
the
n
um
b
er
−
1
solution
of
the
inequation
:
x
2
+
4
x
4
a
b
What
can
w
e
deduce
ab
out
the
studen
t’s
prop
osed
resolution?
2
Solv
e
this
inequation
correctly
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/11722
sacados/11722
3
m
9
m
x
A
B
C
D
E
F
G
chapExoCorrec/11723
sacados/11723
A
B
C
D
E
F
G
H
I
2
x
x
4
cm
4
cm
chapExoCorrec/471
sacados/471
1
1
x
2
1
1
x
·
(1
x
2
·
(1
x
1
2
2
x
−
1
2
x
−
1
2
x
chapExoCorrec/479
sacados/479
13.
F
urther
study
:
systems
of
ine
qualities
E.319
Giv
e
the
v
alue
of
a
so
that
the
system
b
elo
w
has
as
its
solution
the
in
terv
al
1
;
3
:
3
x
+
2
5
1
2
−
x
a
E.318
Three
real
n
um
b
ers
a
,
b
and
c
v
erify
the
follo
wing
system
:
a
·
b
·
c
<
0
a
·
b
>
0
a
+
b
>
0
Find
the
sign
of
these
three
n
um
b
ers.
E.348
Solv
e
the
follo
wing
systems
of
equa-
tions
:
1
2
x
−
3
<
5
x
−
1
x
+
4
3
x
−
2
2
3
x
−
3
>
x
+
1
2
x
+
1
2
x
+
1
14.
De
ep
ening
:
manipulation
of
fr
ames
E.317
Consider
a
AB
C
D
square
with
5
cm
sides.
1
Giv
e
the
exact
length
of
the
diagonals
of
this
square.
2
Kno
wing
that
:
1.41
<
2
<
1.42
Giv
e
a
frame
for
the
length
of
this
diagonal.
E.315
1
Let
a
∈
R
suc
h
that
1
2
<
a
<
1
,
sp
ecify
whic
h
frames
are
v
erified
b
y
a
:
a
1
2
<
1
a
<
2
b
0
<
a
<
2
a
c
1
4
<
a
2
<
4
2
By
case
disjunction
for
a
∈
−
1
;
0
and
a
∈
0
;
2
,
giv
e
a
frame
for
a
2
and
a
3
for
−
1
<
a
<
2
.
15.
Unclassifie
d
exer
cises
E.470
When
solving
an
equation
or
inequal-
it
y
,
the
ˇ
domain
of
solution
ı
ı
is
the
set
of
v
alues
that
the
unkno
wn
x
can
tak
e.
W
e
will
then
lo
ok
for
solutions
among
the
v
alues
in
the
solution
set.
1
Here
is
a
studen
t’s
solution
to
equation
x
2
=
3
x
:
(
E
)
:
x
2
=
3
x
x
2
x
=
3
x
x
x
=
3
F
rom
this,
the
studen
t
concludes
that
the
equation
has
one
solution
:
3
.
a
This
equation
has
a
second
solution.
Whic
h
one?
b
In
their
solution
metho
d,
this
studen
t
used
R
∗
as
the
solution
set
rather
than
R
.
Explain
wh
y?
2
A
studen
t
solv
es
an
inequalit
y
as
follo
ws
:
(
I
)
:
x
+
1
x
2
x
+
1
2
x
1
x
The
studen
t
concludes
that
the
set
of
solutions
is
the
in
terv
al
[1
;
+
∞
[
a
Giv
e
the
domain
of
the
algebraic
expression
x
+1
x
.
b
Sho
w
that
−
1
is
a
solution
to
the
inequalit
y
.
c
What
mistak
e
did
the
studen
t
mak
e?
Implicitly
,
what
solution
set
did
he
or
she
use?
d
Solv
e
the
inequalit
y
(
I
)
using
a
sign
table.
A
dditional
question
:
3
Without
using
sign
tables
:
a
Assume
that
x
is
a
strictly
negativ
e
n
um
b
er
;
solv
e
the
inequalit
y
(
I
)
.
b
Assume
that
x
is
a
strictly
p
ositiv
e
n
um
b
er
;
solv
e
the
inequalit
y
(
I
)
.
c
Use
this
to
deduce
and
find
the
answ
er
to
question
3
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/319
sacados/319
chapExoCorrec/318
sacados/318
chapExoCorrec/348
sacados/348
chapExoCorrec/317
sacados/317
chapExoCorrec/315
sacados/315
chapExoCorrec/470
sacados/470
0
;
60
0
;
80
0
;
40
0
;
20
Image A
0
;
36
0
;
64
0
;
16
0
;
04
Image B
0
;
77
0
;
89
0
;
63
0
;
45
Image C
0
0.5
1
0.5
1
C
f
E.309
The
shortest
distance
b
et
w
een
t
w
o
p
oin
ts
is
a
straigh
t
line.
This
fact
is
expressed
b
y
the
triangle
inequalit
y
relating
the
distances
b
et
w
een
three
p
oin
ts
:
for
an
y
p
oin
ts
A
,
B
,
C
in
the
plane,
the
follo
wing
alw
a
ys
holds
:
AB
AC
+
C
B
(Passing
thr
ough
p
oint
C
lengthens
the
p
ath
unless
it
lies
on
se
gment
)
.
W
e
will
construct
an
isosceles
triangle
D
E
F
at
F
suc
h
that
D
E
=
5
.
Let
x
=
E
F
=
D
F
1
Be
giv
en.
Can
this
triangle
b
e
constructed
for
x
=
2
?
2
Giv
e
a
range
of
p
ossible
v
alues
for
x
.
3
Express
the
v
alue
p
of
the
p
erimeter
of
triangle
D
E
F
in
terms
of
the
v
alue
x
.
4
F
or
x
satisfying
the
range
:
2
;
5
x
5
.
Giv
e
the
corresp
onding
b
ounding
rectangle
for
p
.
E.6755
A
blac
k-and-white
digital
image
is
made
up
of
small
squares
(pixels)
ranging
in
color
from
white
through
all
shades
of
gra
y
to
blac
k.
Eac
h
shade
is
co
ded
b
y
a
real
x
as
follo
ws
:
x
=
0
for
white
;
x
=
1
for
blac
k
;
x
=
0.01
;
x
=
0.02
and
so
on
up
to
x
=
0.99
in
steps
of
0.01
for
all
in
termediate
shades
(fr
om
light
to
dark)
.
The
image
A
,
b
elo
w,
is
comp
osed
of
four
pixels
and
giv
es
a
sample
of
these
shades
with
their
co
des.
Image
retouc
hing
soft
w
are
uses
digital
functions
kno
wn
as
ˇ
re-
touc
hing
functions
ı.
A
function
f
defined
on
the
in
terv
al
0
;
1
is
said
to
ˇ
function
of
retouc
he
ı
if
it
has
the
follo
wing
four
prop
erties
:
f
(0)
=
0
;
f
(1)
=
1
;
f
is
con
tin
uous
on
the
in
terv
al
0
;
1
;
f
is
increasing
on
the
in
terv
al
0
;
1
.
A
shade
co
ded
x
is
said
to
b
e
dark
ened
b
y
the
function
f
if
f
(
x
)
>
x
,
and
ligh
tened,
if
f
(
x
)
<
x
.
if
f
(
x
)
=
x
2
,
a
pixel
of
co
ded
shade
0.2
will
tak
e
on
the
co
ded
shade
0.2
2
=
0.04
.
Image
A
will
b
e
transformed
in
to
image
B
b
elo
w.
If
f
(
x
)
=
x
,
the
co
ded
shade
0.2
will
tak
e
on
the
co
ded
shade
0.2
≈
0.45
.
The
image
A
will
b
e
transformed
in
to
the
image
C
b
elo
w.
Consider
the
function
f
defined
on
the
in
terv
al
0
;
1
b
y
:
f
(
x
)
=
4
x
3
−
6
x
2
+
3
x
W
e
admit
that
the
function
f
is
a
touc
h-up
function.
Its
represen
tativ
e
curv
e
C
f
is
giv
en
b
elo
w
:
1
Graphically
solv
e
the
inequation
f
(
x
)
x
,
using
the
graph
giv
en
b
elo
w,
sho
wing
the
useful
dotted
lines.
2
In
terpret
this
result
in
terms
of
ligh
tening
or
dark
ening.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/309
sacados/309
chapExoCorrec/6755
sacados/6755
0
;
60
0
;
80
0
;
40
0
;
20
Image A
0
;
36
0
;
64
0
;
16
0
;
04
Image B
0
;
77
0
;
89
0
;
63
0
;
45
Image C
0
0.5
1
0.5
1
C
f
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
C
nombre de pièces en milliers
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
milliers d’euros
50
100
150
200
250
300
350
400
450
500
C
R
E.7005
The
compan
y
BBE
(Bio
Bois
Én-
er
gie)
man
ufactures
and
sells
w
o
o
d
p
ellets
to
fuel
b
oilers
and
sto
v
es
in
homes
and
comm
unities.
The
compan
y
pro
duces
b
et
w
een
1
and
15
tons
of
p
ellets
p
er
da
y
.
Daily
man
ufacturing
costs
are
mo
deled
b
y
the
function
C
defined
on
the
in
terv
al
1
;
15
b
y
:
C
(
x
)
=
0.3
·
x
2
−
x
+
e
−
x
+5
où
x
denotes
the
quan
tit
y
of
p
ellets
in
tons
and
C
(
x
)
the
corresp
onding
daily
man
ufacturing
cost
in
h
undreds
of
euros.
In
the
compan
y
BBE
the
selling
price
of
one
ton
of
w
o
o
d
p
ellets
is
300
euros.
The
compan
y’s
daily
rev
en
ue
is
therefore
giv
en
b
y
the
function
R
defined
on
the
in
terv
al
1
;
15
b
y
:
R
(
x
)
=
3
·
x
où
x
denotes
the
quan
tit
y
of
p
ellets
in
tons
and
R
(
x
)
the
corresp
onding
daily
rev
en
ue
in
h
undreds
of
euros.
W
e
define
b
y
D
(
x
)
the
compan
y’s
daily
net
income
in
h
undreds
of
euros,
i.e.
the
difference
b
et
w
een
rev
en
ue
R
(
x
)
and
cost
C
(
x
)
,
où
x
denotes
the
quan
tit
y
of
p
ellets
in
tons.
On
the
graph
b
elo
w,
w
e
giv
e
C
and
Δ
the
resp
ectiv
e
graph-
ical
represen
tations
of
the
functions
C
and
R
in
a
frame
of
reference
with
origin
O
.
The
follo
wing
questions
will
b
e
answ
ered
using
the
graph,
and
with
the
precision
p
ermitted
b
y
it.
No
justification
is
required.
1
Determine
the
quan
tit
y
of
p
ellets
in
tons
for
whic
h
the
compan
y’s
daily
cost
is
minimal.
2
a
Determine
the
v
alues
C
(6)
and
R
(6)
then
deriv
e
an
estimate
of
of
the
daily
net
income
in
euros
generated
b
y
the
compan
y
for
6
tons
of
p
ellets
man
ufactured
and
sold.
b
Determine
the
p
ossible
quan
tities
of
p
ellets
in
tons
that
the
compan
y
m
ust
pro
duce
and
sell
daily
to
generate
a
p
ositiv
e
net
result,
i.e.
a
profit.
E.7055
A
compan
y
pro
duces
and
sells
electronic
comp
onen
ts.
Its
mon
thly
pro
duction
capacit
y
is
b
et
w
een
1
000
and
30
000
pieces.
It
is
assumed
that
all
pro
duction
is
mark
eted.
Giv
en
b
elo
w
are
R
and
C
the
resp
ectiv
e
graphical
represen
ta-
tions
of
the
rev
en
ue
and
cost
functions
on
the
in
terv
al
1
;
30
.
By
graphical
reading,
giv
e
an
estimate
of
the
v
alues
requested.
1
What
is
the
pro
duction
cost
of
21
000
pieces?
2
F
or
what
quan
tities
of
parts
pro
duced,
do
es
the
compan
y
mak
e
a
profit?
3
F
or
what
n
um
b
er
of
pieces
pro
duced
is
the
profit
maxi-
m
um?
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7005
sacados/7005
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
C
chapExoCorrec/7055
sacados/7055
nombre de pièces en milliers
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
milliers d’euros
50
100
150
200
250
300
350
400
450
500
C
R
nombre de jours
0
10
20
30
40
50
60
70
80
milliers d’habitants
2
4
6
8
10
12
E.7060
The
p
opulation
ev
olution
of
a
sea-
side
resort
for
the
summer
2015
w
as
mo
deled
b
y
a
function
f
,
defined
on
the
in
terv
al
0
;
70
,
whose
represen
tativ
e
curv
e
is
giv
en
b
elo
w.
Where
x
is
the
n
um
b
er
of
da
ys
after
1
er
July
,
f
(
x
)
denotes
the
p
opulation
in
thousands.
Th
us,
x
=
30
corresp
onds
to
31
July
and
f
(30)
represen
ts
the
p
opulation
exp
ected
to
b
e
accommo
dated
on
31
July
.
It
is
estimated
that
one
inhabitan
t
will
use
b
et
w
een
45
and
55
liters
of
w
ater
p
er
da
y
.
The
answ
ers
to
the
follo
wing
questions
are
to
b
e
pro
vided
b
y
graphical
reading.
1
a
Estimate
the
maxim
um
n
um
b
er
of
inhabitan
ts
presen
t
in
the
seaside
resort
according
to
this
mo
del
during
the
summer
2015
and
sp
ecify
when
this
maxi-
m
um
w
ould
b
e
reac
hed.
b
The
comm
une
is
able
to
supply
600
000
liters
of
w
ater
p
er
da
y
,
is
this
enough?
2
Estimate
the
n
um
b
er
of
da
ys
on
whic
h
the
n
um
b
er
of
in-
habitan
ts
of
the
resort
is
exp
ected
to
remain
ab
o
v
e
80
%
of
the
maxim
um
n
um
b
er
exp
ected.
E.5661
Complete
the
table
b
elo
w
:
a
b
Compare
a
and
b
Sign
of
b
−
a
5
3
2.7
4
3
7
5
7
4
3
5
6
4
ı
E.6429
1
Giv
e
the
sign
of
eac
h
of
the
follo
wing
calculations
:
a
5
−
3
b
2.4
−
(
−
3.2)
c
3.6
−
7.9
d
2
3
−
5
6
e
6
14
−
9
21
f
3
−
ı
2
Deduce
from
the
previous
question
the
comparison
of
the
follo
wing
n
um
b
ers
:
a
5
:
:
:
3
b
2.4
:
:
:
−
3.2
c
3.6
:
:
:
7.9
d
2
3
:
:
:
5
6
e
6
14
:
:
:
9
21
f
3
:
:
:
ı
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7060
sacados/7060
Extrait Antilles-Guyane Septembre 2015
nombre de jours
0
10
20
30
40
50
60
70
80
milliers d’habitants
2
4
6
8
10
12
chapExoCorrec/5661
sacados/5661
chapExoCorrec/6429
sacados/6429
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