a: (x-3)(x-2)<0
=>x-2>0 và x-3<0
=>2<x<3
b: \(\left(x+3\right)\left(x+4\right)\left(x^2+2\right)\ge0\)
=>(x+3)(x+4)>=0
=>x+3>=0 hoặc x+4<=0
=>x>=-3 hoặc x<=-4
c: \(\dfrac{x-1}{x-2}\ge0\)
=>x-2>0 hoặc x-1<=0
=>x>2 hoặc x<=1
d: \(\dfrac{x+3}{2-x}>=0\)
=>\(\dfrac{x+3}{x-2}< =0\)
=>x+3>=0 và x-2<0
=>-3<=x<2
Giai cac bất phương trình:
a) (x-3)(x-2)<0
b) (x+3)(x+4)(x2+2)\(\ge\) 0
c) \(\dfrac{x-1}{x-2}\) \(\ge\)0
d)\(\dfrac{x+3}{2-x}\)\(\ge\) 0
e) (x-3)(x-2)(x+1)<0
g) \(\dfrac{2}{x-1}\)<0
k) x2 +3x+2>0
m) x2+1<0
Bài 2 ) Giai phương trình
a)\(\)/x-1/=-3
b) /2x+1/=0
c)/3-2x/=4
d)/x+1/+3x=4
e) /x+1-4x/=5
g) /x-1/+/x-3/=2
k)/x-1/+/x-2/+/x-3/=2
Chứng minh các bất đẳng thức:
a) \(x^2+y^2\ge\dfrac{\left(x+y\right)^2}{2}\ge2xy\)
b) \(\dfrac{1}{x}+\dfrac{1}{y}\ge\dfrac{4}{x+y}\) với \(x>0,y>0\)
Câu 1: Giải PT:
a) 2x2 - 6x + 1 = 0
b) x3 + x = 2
c) (x-2)(x+1) < 0
d) \(\dfrac{2x-5}{x+5}\) > 0
Câu 2: Chứng minh bất đẳng thức sau:
a) 2x - x2 \(\le\) 1 với mọi x
b) A = (a+b)\(\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\)\(\ge\) 4
c) B = \(\dfrac{a+b}{c}+\dfrac{b+a}{a}+\dfrac{c+a}{b}\ge6\) (a,b,c > 0)
d) \(\dfrac{a}{4b^2+1}+\dfrac{b}{4a^2+1}\ge\dfrac{1}{2}\) (a,b dương; a+b=4ab)
Bài 1: Giải các phương trình và bất phương trình sau:
a. \(5x-10=0\)
b. \(\dfrac{2}{x+1}-\dfrac{3}{x-2}=\dfrac{2x-6}{\left(x+1\right)\left(x-2\right)}\)
c. \(3x-5\ge-7\)
d. 3x - 1 = 0
Tìm x, biết:
a) 2x+ 5 < 3
b) 2 - x ≥ 5 - 2x
c) -x - 3 > 0
d) 2x-5(x+1) < x - 8
e) 4x-2 > 5x+1
f) -2x+ 4 ≥ 0
g) 8x + 3 ≤ 5x- 1
a/ Với x>0, CM: x+\(\dfrac{1}{x}\)\(\ge\)2
b/ Cho a,b,c là 3 số dương, cm: \(\dfrac{a^2}{b+c}+\dfrac{b^2}{a+c}+\dfrac{c^2}{a+b}\ge\dfrac{a+b+c}{2}\)
a \(x^2-x=0\) b \(x^2-2x=0\) c (x+1)(x+2)=(2-x)(x+2)
d \(\dfrac{1}{x-2}+3=\dfrac{3-x}{x-2}\) đ \(\dfrac{8-x}{x-7}-8=\dfrac{1}{x-7}\)
e \(\dfrac{x}{2\left(x-3\right)}+\dfrac{x}{2\left(x+1\right)}=\dfrac{2x}{\left(x+1\right)\left(x-3\right)}\)
f \(5+\dfrac{76}{x^2-16}=\dfrac{2x-1}{x+4}-\dfrac{3x-1}{4-x}\)
g \(\dfrac{90}{x}-\dfrac{36}{x-6}=2\) h \(\dfrac{1}{x}+\dfrac{1}{x+10}=\dfrac{1}{12}\) i \(\dfrac{x+3}{x-3}-\dfrac{1}{x}=\dfrac{3}{x\left(x-3\right)}\)
k \(\dfrac{3}{x+2}-\dfrac{2}{x-2}+\dfrac{8}{x^2-4}=0\) l \(\dfrac{3}{x+2}-\dfrac{2}{x-3}=\dfrac{8}{\left(x-3\right)\left(x+2\right)}\)
m\(\dfrac{x}{2x+6}-\dfrac{x}{2x+2}=\dfrac{3x+2}{\left(x+1\right)\left(x+3\right)}\)
n \(\dfrac{x}{x+1}-\dfrac{2x-3}{1-x}=\dfrac{3x^2+5}{x^2-1}\) j \(\dfrac{5}{x+7}+\dfrac{8}{2x+14}=\dfrac{3}{2}\)
q \(\dfrac{x-1}{x}-\dfrac{1}{x+1}=\dfrac{2x-1}{x^2+x}\)
Cần gấp ạ
A=\(\dfrac{x^2+3}{x^2-9}\) + \(\dfrac{2}{x+3}\) - \(\dfrac{1}{x-3}\) (x \(\ge\) 0 ; x \(\ne\) 9)
a.Rút gọn A
b.Tìm x, A= \(\dfrac{x+1}{x+2}\)
M= \(\left(\dfrac{x^2+x+10}{x^2-9}+\dfrac{1}{x-3}\right)\): \(\dfrac{1}{x-3}\) (x \(\ge\) 0 ; x \(\ne\) 9)
a.Rút gọn M
b.Tìm x để M < x + 1
