已知函数f(x)=2x-1a．（1）解关于x的不等式f（x）＞0；（2）若f（x_爱下问搜题

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问题解答题

已知函数f(x)=

2

x

-

1

a

．
（1）解关于x的不等式f（x）＞0；
（2）若f（x）+2x≥0在（0，+∞）上恒成立，求实数a的取值范围．

答案

（1）不等式f（x）＞0，即

2

x

-

1

a

＞0

整理，得

2a-x

ax

＞0，等价于ax（x-2a）＜0

因为a≠0，可得

①a＞0时，解之得0＜x＜2a；②a＜0时，等价于x（x-2a）＞0，解之得x＜2a或x＞0

综上所述，得：

当a＞0时，原不等式的解集为（0，2a）；a＜0时，原不等式的解集为（-∞，2a）∪（0，+∞）．

（2）f（x）+2x≥0在（0，+∞）上恒成立，即

2

x

+2x-

1

a

≥0在（0，+∞）上恒成立，整理得：

2

x

+2x≥

1

a

根据基本不等式，得

2

x

+2x≥2

2

x

•2x

=4

∴不等式

2

x

+2x≥

1

a

（0，+∞）上恒成立，即4≥

1

a

，解之得a＜0或a≥

1

4

．

综上所述，得a的取值范围为（-∞，0）∪[

1

4

，+∞）

选择题

NaNO₂是一种食品添加剂，它能致癌。酸性KMnO₄溶液与NaNO₂的反应方程式是（未

配平）：MnO₄^-＋NO₂^-＋→ Mn²⁺＋NO₃^-＋H₂O。下列叙述中正确的是（）

A．该反应中NO₂^-是还原剂

B．反应过程中溶液的pH 减小

C．生成1 mol NaNO₃时消耗2.5 mol KMnO₄

D．中的粒子是OH^－

完形填空

Lightning flashed through the darkness over Sibson’s bedroom skylight（天窗）.Sibson was shaken by a clap of thunder ____21___ he knew what was happening. The storm had moved directly _____22____ his two-story wooden house. Then he heard the smoke alarm beeping.
Sibson rushed down the stairs barefoot to     23     ; he opened the door to the basement（地下室）, and flames      24     out. Sibson ran back upstairs to call 911 from his bedroom. “I felt      25      because the room had a separate outdoor stairway,” he explains.
But the phone didn’t work, and when he tried to go down the outdoor stairway, he was      26     by a wall of flames. Sibson realized he was trapped（困住）.
Sibson’s house was three kilometers      27     the main road and was so well hidden by trees that he knew calling for help would be     28     .
Up a hill nearby lived Sibson’s neighbor, Huggons. He was lying in bed when something like a smoke alarm      29     his ears. He jumped out of bed, took his     30     and flashlight, and headed down the hillside toward the    31    . That was when he saw the rolling heavy smoke.
Huggons dialed 911, and the operator warned him not to    32   the house. But Huggons said, “There is no way I am going to listen to Sibson    33    and die in that fire.”
“Anyone there?” Huggons called out. Then he heard “Help! I’m trapped!” coming from the second floor balcony（阳台）. He entered the house, but soon had to run back to catch his     34    .
After one more    35   inside the house, Huggons gave up and     36   around back.
The wind parted the smoke just     37    for him to catch sight of Sibson. But there was no way to get to him. He    38   the flashlight into the woods and noticed a ladder. He took it over to the balcony and     39    Sibson down just as the second floor of the house fell off.
Sibson is still     40    when he tells the story. “ I was alone that night,” he says. “Then I heard the most beautiful sound in my life. It was Huggons.”