3
votes
2answers
67 views

An inequality I am stuck on

This is somehow related to this problem but I don't have any idea about it. $a,b,c,d$ are positive reals such that $a+b+c+d=4$ $$\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}+\frac{1}{d+3}\le ...
0
votes
4answers
68 views

Questions about solving inequality: $2 < \frac{3x+1}{2x+4}$

Solve the inequality: $2 < \frac{3x+1}{2x+4}$ Step 1: I simplified $\frac{3x+1}{2x+4}$ into: $3x+1-2x-4= x-3$. Step 2: $2>x-3$ Here I subtracted $2$ from both sides into: $x>-5$ or ...
1
vote
2answers
51 views

Inequality - Find what value of $t$ satisfies: $ (t/24) - (t+1) + (3t/8) < (5/12) (t+1)$

Inequality - Find what value of $t$ satisfies: $(t/24) - (t+1) + (3t/8) < (5/12) (t+1)$. Step 1: I multiplied both sides by $24$ and divided to get: $t-24(t+1)+9t < 10+24(t+1)$. Step 2: I ...
1
vote
1answer
40 views

Find value of $x$ for: $(1/3)(1-x) \geq 2(x-3)$

Find what value of $x$ satisfy: $(1/3)(1-x) \geq 2(x-3)$ First I multiplied both sides by $3$ so that $1/3$ became $3/3=1$. So I tried to find $x$ this way: $(1-x) \geq 6(x-3)$. I tried solving it ...
-8
votes
2answers
60 views

How to solve an irrational inequality?

How to solve the following inequality: $$\sqrt{1-2x} < \sqrt{4 - x}$$ I don't understand why "$(1-2x)$ have to be $\ge 0$". If it was the rule for numbers inside a square root, I was checking ...
2
votes
3answers
110 views

Proving that one of $a(1-b), b(1-c), c(1-a) \le \frac{1}{4}$

how can a prove that at least one of those is less than or equal to 1/4. $$\forall a,b,c\in \mathbb R^+, \ a(1-b)\leq 1/4 \lor b(1-c) \leq 1/4 \lor c(1-a) \leq 1/4.$$ help please!
4
votes
3answers
401 views

An inequality in numbers

Which number is larger? $\underbrace{888\cdots8}_\text{19 digits}\times\underbrace{333\cdots3}_\text{68 digits}$ or $\underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots67}_\text{68 ...
2
votes
3answers
163 views

How can this equality be established by elementary algebraic means?

Let $x \geq 1$. Then is it true that $2x^3 - 3x^2 + 2 \geq 1$? If so, how can I show this using only elementary ideas such as factorisation? Of course, I can demonstrate this using the methods of ...
3
votes
3answers
126 views

Find the maximum value of $xy^2z^3$ given that $x^2 + {y}^2 + {z}^2 = 1$, using AM-GM

I've been struggling with this equation and how to find the maximum value it can take: Maximise $xy^2z^3$ given that $x^2+y^2+z^2 = 1$ The question is from the book Introduction to Inequalities ...
1
vote
1answer
32 views

Solve this with CBS

How can you see the mínimum value of $ 1/x + 4/y + 9/z $ with x+y+z=1 using the CBS inquality? I have seen a proof of that that use trigonometric substitutions, but i don´t see as one-step the ...
4
votes
2answers
46 views

If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5>\ldots?$

Problem : If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5$ is always greater than : (a) $\ 7 $ (b) $\ 8 $ (c) $\ 9 $ (d) $\ $none of these Solution : We ...
0
votes
2answers
42 views

Is $3(2k+1)(2^{2k+1}-1)>(2^{k+3}-1)(2^{k+1}-1)$?

Let $k$ be an integer. I need to prove that: $$3(2k+1)(2^{2k+1}-1)>(2^{k+3}-1)(2^{k+2}-1)$$ where $k>a$ for a suitable $a$. thanks in advance.
3
votes
4answers
237 views

inequality method of solution

Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$ I've tried first determining when the absolute value ...
0
votes
3answers
35 views

Simplifying and understanding inequalities in two variables

given an equation: $$\frac{x-y}{x+y}\ge0$$ What are the steps to simplify this into an understandable group of inequalities which will yield a solution set as a group of areas? I already know that ...
1
vote
2answers
32 views

What will be the range of $f(x)= \frac{12}{\sqrt{(15-2x-x^2)}}$

Here's my try: Since the denominator involves a square root so I solved the following inequality: $15-2x-x^2>0$ which gives a solution set of $x=(-5,3)$. This is the domain of $f(x)$. However since ...
-1
votes
2answers
122 views

How to show $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$?

I was trying to solve $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$ and I keep getting a partial answer of $x>4$ though answer key suggests a more expanded ...
0
votes
6answers
68 views

solving the inequalty

are there any ways to solve :$ x^4 -6x^3 +28x^2 -64x +96 >0$ ?
0
votes
1answer
48 views

Linear Programming?

An agriculture company has 80 tons of fertilizer Alpha and 120 tons of fertilizer Bravo. The company mixes these fertilizer into two products. Product Super requires 2 parts of fertilizer A and 1 part ...
2
votes
1answer
41 views

Solve: $\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$

Given $a_1,a_2,\ldots,a_n \in\mathbb{R}$. Solve the following equation on $\mathbb{R}$: $$\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$$ I am not sure that a closed-form solution exists, so iterative ...
0
votes
1answer
34 views

$f(x)=sec(x)$ inequality inconsistency\trouble

I'm currently attempting to find the range of $f(x)=\sec(x)$ by considering $\cos(x)$ in the intervals of $0<\cos(x)\leqslant 1$ and $-1\leqslant \cos(x)<0$ (as $\sec(x)$ is undefined for ...
7
votes
3answers
192 views

Is $ln(x)$ ever greater than $x$

Is $\forall x \in \mathbb{R}, \ln(x) \lt x$ a true statement? Just wondering for some convergence related thing
3
votes
2answers
87 views

QM-AM-GM-HM proof help

Out of interest, I am trying to proof QM-AM-GM-HM inequality. If you don't know it, it's something like this... Let there be $n$ numbers $x_1, x_2, x_3...x_n$, where $x_1, x_2, ...,x_n>0$. Proof ...
0
votes
2answers
30 views

Equations,inequalities and absolute values.

I really am confused when I am supposed to change the $<$ and $>$ symbol. For example this unsolved question in my reference book: solve $\displaystyle \frac{x+1}{x-1}>0$ One ...
-3
votes
4answers
77 views

How to solve:$\frac{3}{x-5}< \frac{7}{x-2}$

How to solve: $\dfrac{3}{x-5}< \dfrac{7}{x-2}$ Any smart ideas for this one ? Could I rewrite this as $\dfrac{x-5}{3}> \dfrac{x-2}{7}$ ?
2
votes
2answers
41 views

Inequality involving multiple square roots

Wolfram alpha solves $\sqrt{x+1}\ge\sqrt{x+2}+\sqrt{x+3}$ for $x$, and answers $x=-2/3(3+\sqrt{3})$. How did it do it? Thanks!
3
votes
5answers
138 views

How to solve: $\frac{3·x-5}{8·x-2}<6$

I'm trying to solve $\frac{3x-5}{8x-2}<6$ ? I'm not sure which first step to take. I mean if I multiply both sides by $8x-2$ then I'm not sure if the sign would switch, as this could be positive ...
-2
votes
1answer
40 views

Comparing floor and ceiling fractions

Is the following true for all integers x>1: $\lfloor{\frac{2x}{3}}\rfloor \geq \lceil \frac{x}{2}\rceil$
4
votes
0answers
60 views

Showing that $|x-y| \leq |x| +|y|$ for $x.y \in \mathbb{R}$.

I know from intuition that $|x-y| \leq |x| +|y|$ for $x.y \in \mathbb{R}$. The way I would prove it is to use the triangle inequality: $|x-y| = |x+(-y)| \leq |x| +|-y| = |x|+|y|$ for $x.y \in ...
0
votes
1answer
34 views

How many equations of this inequality?

What is the other equation for this inequality? $|2x - 7| \ge 1$ Is it $|2x - 7| \ge -1$ or $|2x - 7| \le -1$
1
vote
5answers
49 views

Algebra - solve given inequality

I am having problems understanding how to solve: $ x^2 - x - 1 > 0 $. Any help would be much appreciated.
0
votes
1answer
35 views

algebra - Solve given innequality

I am having problems understanding how to solve: $$ x^2 -5x \geqslant 0 $$ Any help would be much appreciated as I have been stuck on this question for quite some time.
5
votes
3answers
222 views

Need algebra tip about $a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$ for sides of a triangle

I just got a long expression: $$a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$$ and I need to prove its less than zero for every $a$, $b$, and $c$ which are triangle sides I really need tips how to ...
1
vote
1answer
40 views

Quadratic inequality with parameter

Hi I've got this inequality with parameter $a\in R$ $\frac{x+a}{x}\le x+2$ I've solved it but I've got different results than book. I've done it by dividing it into 2 cases. 1. x<0 2. x>0 and then ...
6
votes
2answers
131 views

How to arrange $e^3,3^e,e^{\pi},\pi^e,3^{\pi},\pi^3$ in the increasing order?

For these six numbers, $e^3,3^e,e^{\pi},\pi^e,3^{\pi},\pi^3$, how to arrange them in the increasing order? This problem is taken from the today test: National Higher Education Entrance Examination. ...
6
votes
1answer
81 views

Prove that ${1\over2}<{1\over1001}+…+{1\over2000}<1$

Prove that ${1\over2}<{1\over1001}+......+{1\over2000}<1$ Can it be proved by langrange's mean value theorem or by convert it into a Riemann sum?
-2
votes
2answers
79 views

Verify if this inequality is true

If $a_1, \; a_2, \cdots , \; a_n$ are positive real numbers with product $1$, does this inequality hold, and if so how can one prove it? $$(\sum a_i)^{7} \geq n^{5} \cdot (\sum a_i^2)^2$$ Thanks ...
-2
votes
5answers
61 views

Where is $(x-1)(x+1/2)\geq0$

The solution is $x\geq 1$ or $x\leq-(1/2)$ However, I do not understand why you flip the sign of $-1/2$ but not $1$.
1
vote
2answers
19 views

Quadratic inequalities involving two solutions

I have a quadratic expression, which I have factored to correctly be: $(x-9)(x-2) > 0$ However, I don't know how to determine the two values of X after this, the correct answers are x < 2, x > ...
1
vote
2answers
120 views

Inequality manipulation in two variables

Given the inequality $m^4/s^5 - m \leq 2s^2$, the paper I am reading says that this implies that $s \geq 1/10\cdot m^{4/7}$. I tried manipulating the inequality to try and reproduce their result, but ...
0
votes
2answers
23 views

How is this algebraic step justified. (Inequalities)

I don't understand why this is allowed or the logic behind it: $P[X^2 - 2X < 8] = P[x^2 -2X + 1 < 9] = p[ (X - 1)^2 < 9 ] $ $P[-3 < (X - 1) < 3]$ (this step right here). What is the ...
4
votes
1answer
76 views

Prove $\displaystyle\frac{a}{b+3}+\frac{b}{c+3}+\frac{c}{d+3}+\frac{d}{a+3}\le 1$. [duplicate]

Given $a,b,c,d\ge 0$ and $a^2+b^2+c^2+d^2=4$ show the following holds : $$\displaystyle\frac{a}{b+3}+\frac{b}{c+3}+\frac{c}{d+3}+\frac{d}{a+3}\le 1$$ Now I tried to fully expand but that becomes too ...
3
votes
1answer
88 views

solve the inequality: $\displaystyle \ln|x^2 -3x+2|+\frac{2x-3}{x-2} \geq 0$

How can I solve the following inequality in the set of real numbers: $\displaystyle \ln|x^2 -3x+2|+\frac{2x-3}{x-2} \geq 0$ Thanks in advace!
1
vote
4answers
71 views

Solving the inequality. ${3x + 5 \over x} \gt 0$

I have a question similar to this: $${3x + 5 \over x} \gt 0$$ I am not sure why but the 0 is throwing me off as I want to $0 \over x$ to balance it and Ill just get 0 again and just seems wrong.
0
votes
1answer
33 views

A simpler expression that is always smaller or larger

I have this function $$\small A(n)=-\frac{2 b_0 \left(2 - 2 l (-2 + n) - 3 l^2 (-1 + n) + l^3 (-1 + n)^2 \right) + l \left(2 + 3 l - l^2 (-2 + n) \right) \sigma^2 \log\left(\frac{q}{1-q}\right)}{ ...
2
votes
4answers
319 views

How to solve this weird inequality?

$\frac{x-1}{x+1} < x$ Thanks! I did the following. $\frac{x-1}{x+1} - x< 0 /-x$ $\frac{x-1 - x(x+1)}{x+1} < 0$ $\frac{-x^2-1}{x+1} < 0$ What to do next?
3
votes
1answer
83 views

How prove $((x-y)(y-z)(z-x))^2\le 2((x^2-y^2)^2+(y^2-z^2)^2+(z^2-x^2)^2)$

let $x,y,z>0$, and such $$x^2+y^2+z^2=x^2y^2+y^2z^2+x^2z^2$$ show that $$((x-y)(y-z)(z-x))^2\le 2((x^2-y^2)^2+(y^2-z^2)^2+(z^2-x^2)^2)$$ My try: let $$x-y=a,y-z=b,z-x=c\Longrightarrow a+b+c=0$$ ...
1
vote
3answers
51 views

Inequality involving conjugate numerator/denominator pairs

Question is to solve: $$\frac{(x-2)(x-4)(x-7)}{(x+2)(x+4)(x+7)} > 1$$ I thought I could negate terms to make them equal (i.e. $-(x-2)$), but that does not happen. I could subtract $1$ from ...
2
votes
1answer
86 views

Prove that $(n!)^{2}>(n)^n$ using inequality [duplicate]

To prove that $(n!)^2>(n)^n$ That is to prove that $(n!)^2-(n)^n>0$ Now, $(n!)(n!)-(n)^n=[n.(n-1)(n-2)...][n.(n-1)(n-2)...]-[n.n.n...]$ $[n^2.(n-1)^2 . (n-2)^2 ...]-[n.n.n...]$ Comparing ...
0
votes
3answers
40 views

Inequalities - what am I doing wrong

I have the inequality: $$(14x)/(5-x) \le 7x$$ I tried solving it this way: $$ 14x \le 7x(5-x) \\ 2 \le (5-x) \\ -3 \le -x \\ x \le 3 $$ Apparently, the answer is $$0\le x\le 3$$ What am I doing ...
2
votes
3answers
44 views

Solving this inequality

Question: Solve: $$\frac{5x-6}{x+6}<1$$ My attempt: $$\frac{5x-6-x-6}{x+6}<0$$ $$\Rightarrow \frac{4x-12}{x+6}<0$$ $$\Rightarrow \frac{x-3}{x+6}<0$$ $$\Rightarrow (x-3)(x+6) < ...