# Tagged Questions

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### Show that $ax^2+2hxy+by^2$ is positive definite when $h^2<ab$

The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$." Now ...
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### The process of solving the inequality $\frac{8}{19} x\ge -1$

Why did he multiply both sides by 19/8 and not 8/19 ? Is this a rule when dealing with inequalities that to remove fractions, you have to multiply by the reciprocal ?
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### How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
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### Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which syas what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
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### Arithmetic and Geometric Mean Inequalities [on hold]

Can someone help me to understand the logic of: $$\sqrt{ab} \le \frac{a+b}{2}$$ Proof: ?
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### If $a,b,c$ are positive, then $(a+b+c)(1/a+1/b+1/c)\ge 9$

The question asks to prove that if "$x_1,x_2,x_3$ are positive numbers show that: $$(x_1+x_2+x_3) \left(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3} \right)\ge 9$$ I've tried to use the fact that the ...
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### Largest number of pairs that can be added while keeping the population at least 60% male

I'm doing problems from the AoPS Algebra Beginner's book. There's this problem that states the following, At her ranch, Georgia starts an animal shelter to save dogs. After the first three days, she ...
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### Can the inequality $a^3 + b^3 + c^3 \ge a^2b + ac^3 + b^2c$ be derived from arithmetic-geometric means? [duplicate]

The inequality goes as follow: $$a^3 + b^3 + c^3 \ge a^2b + ac^3 + b^2c$$ Where $a,b,$ and $c$ are positive real numbers. Also, can it be solved using am-gm?
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### Minimizing the expression $(1+1/x)(1+m/y)$ over positive reals such that $mx+y=1$

Let $x$ and $y$ be positive real numbers such that $mx+y=1$. Find the positive $m$ such that the minimum of: $$\left( 1 + \frac{1}{x} \right)\left( 1 + \frac{m}{y} \right).$$ is $81$. I have ...
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### Why do the relations $ab=1/2$ and $a>b$ imply $a^2>1/2>b^2$ for positive $a,b$?

When I was reading a probstat book, I encountered an example which I am able to understand except for a formula which I am not able to grasp. It may be basic but I am not able to get it, the solution ...
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### How to solve the inequality $x^2>10$ using square roots?

Solve the inequality: $$x^2>10$$ How am I supposed to do this? It doesn't make sense when I take into account that if $x^2=10$ then $x=+\sqrt{10}$ and $x=-\sqrt{10}$ But how am I supposed to ...
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### Solving the logarithimic inequality $\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$

I tried solving the logarithmic inequality: $$\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$$ several times but keeping getting wrong answers.
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### inequality funny question

I'm not sure what they want here: solve the inequality in realtion to $x$ for various values of $a$ : $\frac{(a+2)x}{a-1} - \frac{2}{3} < 2x-1$
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### Solve the inequality $(1/2)^x-(1/2)^{-1-x}\ge1$ for real $x$

I have to solve in $\Bbb{R}$ the following inequality : $$\left(\frac{1}{2}\right)^{x} - \left(\frac{1}{2}\right)^{-1 - x} \ge 1 \qquad(E)$$ So far I have : For $x=0$ this inequality if not ...
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### Inequality in four variables which sum up to 4

The positive real numbers $x,y,z,t$ satisfy $x+y+z+t=4$. Is the inequality $$x\sqrt{y}+y\sqrt{z}+z\sqrt{t}+t\sqrt{x}\leq4$$ true for all $x,y,z,t>0$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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!