51
Tu 0≤–a<4, anu –4<a≤0, maSin utolobaTa sistemas
aqvs erTi amonaxsni: x=0.
Tu a=–4, sistemas mxolod ori amonaxsni aqvs,
x=0 da
x=4.
Tu –a>4 an –a<0, sistemas uamravi amonaxsni aqvs.
kerZod, –a>4 SemTxvevaSi x∈[4; –a]∪{0} (ixileT su-
raTi). –a<0 SemTxvevaSi x∈[–a; 0].
pasuxi: a) aseTi
a ar arsebobs, b) Tu –4<
a≤0, maSin aqvs mxolod erTi amonaxsni
x=0, g) aqvs uamravi amonaxsni, Tu
a<–4 (maSin
x∈[4; –
a]∪{0}) an
a>0 (maSin
x∈[–
a; 0]).
21
milis ganivi kveTis radiusia (santimetrebSi) d – 0,4
2 . pirobiT,
5000≤π
(
d – 0,4
2
)
2
≤
10000 (yvela monacemi sm-ebSia).
saidanac, Tu SemovitanT
d – 0,4
2 =t aRniSvnas, miviRebT
Z
[
\
]
]
]
]
t
2
≤
3184,71
t
2
≥
1592,36
(miaxloebiT).
t>0 pirobis gaTvaliswinebiT,
39,9<t<56,4,
80,2<d<113,2.
22
ferdi aRvniSnoT 2x-iT, maSin pirobiT,
10≤ 5 + 2x
2 ⋅x
√
3 ≤15,
2
x
2
⋅
√
3 +5x
√
3 –30≤0
20≤5
x
√
3 +2x
2
⋅
√
3 ≤30,
2x
2
⋅
√
3
+5
x
√
3 –20≥0.
gamovsaxoT am utolobebis amonaxsnebis simravleebi RerZze (miaxloebiT)
x>0 pirobis gaTvaliswinebiT miviRebT:
x∈(1,23
; 1,95),
h=x
√
3 ,
h∈(2,13
; 3,29)
. aq SeiZleba `uxeSi~ miaxloebac. magaliTad,
h∈(2,2; 3,2).
23
Tu navis sakuTar siCqares aRvniSnavT x-iT, dinebis mimarTulebiT siCqare
iqneba x+1, sawinaaRmdegod x–1. pirobiT, 3≤ 10
x+1+
6
x–1 ≤ 4.
gaviTvaliswinoT, rom x+1 da x–1 dadebiTi sidideebia da gavamartivoT utoloba:
3x
2
–3≤16x–4≤4x
2
–4.
–
a
–a
52
es ormagi utoloba tolfasia
Z
[
\
]
]
]
]
4x
2
–16x≥0
3x
2
–16x+1≤0 sistemis, saidanac miaxloebiT x∈[4; 5,25].
21-23, 25, 26 amocanebSi damrgvalebis wesebis garda viTvaliswinebT _
amocanis
pirobebSi metobiTaa damrgvaleba gamarTlebuli, Tu naklebobiT.
24
a) gavixsenoT: Tu
√
a
2
=a, maSin a≥0. mocemul gantolebas akmayofilebs x-is
nebismieri mniSvneloba, romlisTvisac
x
2
–9≥0, x∈(–∞; –3]∪[3; +∞).
b) –
x
2
+7x–6≥0, x
2
–7x+6≤0, saidanac x∈[1;6].
25
a) vTqvaT, turistebi dRiurad xkm-is gavlas apirebdnen. amocanis maTemati-
kuri modeli aseTia
Z
[
\
]
]
]
]
6(x+5)>90
8(x–5)<90, saidanac 10<x<16,25.
pasuxi:
turistebi apirebdnen dReSi 10 km-ze metis da 16,25 km-ze naklebis gavlas.
26
moednis sigrZe da sigane aRvniSnoT 5x da 3x-iT. pirobiT, 375≤15x
2
≤
500,
Z
[
\
]
]
]
]
x
2
≤
3313
x
2
≥
25, saidanac
x>0 pirobis gaTvaliswinebiT miviRebT 5≤
x≤5,7 (miaxloebiT).
pasuxi: moednis zomebia (3
x)
da (5
x), sadac
x∈[5; 5,7].
jgufuri muSaoba
Tema:
amocanebis amoxsna sinjvis xerxiT da kvadratuli utolobis SedgeniT.
jgufebs daurigebT dasaxelebul amocanebs da SesTavazebT amoxsnis xerxebs.
amoxsnis yoveli xerxis demonstraciisTvis jgufis wevrebi SeafaseT saTanado
qulebiT. amocanebis amoxsnis prezentaciisa da ganxilvis Semdeg saWiroa Catardes
msjeloba ama Tu im xerxis upiratesobaze mocemul konkretul SemTxvevaSi.
jgufuri muSaobis warmatebiT warmarTvisTvis moswavles unda SeeZlos: ricxvis
Cawera Tanrigis mixedviT (xy=10x+y), asoiTi gamosaxulebis gamartiveba, piTagoras
Teoremis gamoyeneba, marTkuTxedis farTobis gamoTvla, kvadratuli utolobis da
utolobaTa sistemis amoxsna.
a) vTqvaT, saZiebeli orniSna ricxvia xy=10x+y. pirobiT
Z
[
\
]
]
]
]
(10x+y)(10y+x)<1620
10x+y+10y+x>80,
sadac y=x+4. miviRebT:
Z
[
\
]
]
]
]
(11x+4)(11x+40)<1620
22
x>36
,
saidanac miaxloebiT x∈(1,6; 2,8). x cifria, amitom x=2, y=6.
pasuxi: 26.