1
vote
4answers
48 views

Solve the inequality: $|x^2 − 4| < 2$

This is a question on a calculus assignment our class received, I am a little confused on a few parts to the solution, can someone clear a few things up with it? Since $x^2-4 = 0$ that means $x = 2$ ...
0
votes
2answers
59 views

Writing a proof of an inequality between fractions

I have no idea how to do this. Suppose $x,y,z,n$ are positive integers. Given that $\frac{x}{y} < \frac{z}{n}$, prove that $$\frac{x}{y} < \frac{x+z}{y+n} $$
0
votes
1answer
42 views

inequalities and their solutions

a. Since: 9|x + 9| + 6 > 5 9|x + 9| > -1 |x + 9| > -1/9 => always true, every real x value is a solution so it has infinitely many solutions. (−∞, ∞) How would I graph this solution set on a ...
1
vote
2answers
38 views

Turning $2\le x$ into $\sqrt{1+\frac{4}{x^6}}\le \sqrt{5}$?

I am supposed to turn $2\le x$ into $\sqrt{1+\frac{4}{x^6}}\le \sqrt{5}$, and I have no idea on how to approach this. I'll post my steps, even though I don't think they'll be of much help. $$2\le x ...
2
votes
3answers
56 views

Inequality: $2(p^2+q^2+r^2)+2(pq+qr+rp)\ge pqr$

I need to determine the range of $p,q,r$ such that $2(p^2+q^2+r^2)+2(pq+qr+rp)\ge pqr$. I am not given any other information except that $p,q,r\in \mathbb{R}$. I haven't solved a problem like this ...
1
vote
1answer
30 views

An inequality related to Pythagorean theorem: if $A^{2} + B^{2} = C^{2}$, then $A+B>C$

If $A^{2} + B^{2} = C^{2}$, prove $A+B>C$ for all $A>0$ and $B>0$ Intuitively it seems to apply to all positive real numbers(since the hypotenuse of a right triangle is shorter than the sum ...
0
votes
8answers
123 views

Inequality: $x^2+y^2+xy\ge 0$

I want to prove that $x^2+y^2+xy\ge 0$ for all $x,y\in \mathbb{R}$. My "proof": Suppose wlog that $x\ge y$, so $x^2\cdot x\ge x^2\cdot y\ge y^2\cdot y=y^3$ (because $x^2\ge 0$ so we can multiply ...
2
votes
2answers
28 views

Require help with Inequality problems

I am unable to find the solution for below Inequality problems. 1) $2/x<3$ The answer seems to be x belong to $(-\infty,0)\cup (2/3,\infty)$ 2) $\dfrac{x+4}{x-3}<2$ The answer seems to be x ...
1
vote
3answers
63 views

Basic Algebra Inequality Proof

Is there a formal proof that $x<2 \iff -x>-2$, or is this just a matter of convention?
1
vote
2answers
48 views

Inequality on product of two positive numbers

This question is linked to this other question about a proof of the AM-GM inequality. All that I still don't understand, because I don't know how to prove, is the Lemma, by which if $a, b, c \in ...
1
vote
3answers
36 views

What are the steps to solving |3x + 1| > |2x - 7| with the given answer as $(-∞,-8)\cup(6/5,∞)$?

What are the steps to solving $|3x + 1| > |2x - 7|$ with the given answer as $(-∞,-8)\cup(6/5,∞)$? I am having difficulty with understanding inequalities with absolute value functions on both ...
0
votes
1answer
55 views

$\frac{x}{y} \ge \frac{a_1}{b_1} \ge \frac{a_2}{b_2}$ and $b_1 \le b_2 \implies \frac{x+a_1}{y + b_1} \ge \frac{x+a_2}{y + b_2}$?

Given $\frac{x}{y} \ge \frac{a_1}{b_1} \ge \frac{a_2}{b_2}$, where $x,y,a_i,b_i$ are positive numbers. I would like to prove the following: Claim: If $b_1 \le b_2$, then $\frac{x+a_1}{y + b_1} ...
1
vote
1answer
35 views

Trigonometric inequality question [closed]

Let $0 < A < \frac {\pi}{2}$ and $0 < B < \frac {\pi}{2}$. (a) prove that $\sec^2 A + \csc^2 A \cdot \csc^2 B \cdot \sec^2 B \geq 9.$ (b) determine values of $\sec A$ and $\sec B$ when ...
0
votes
1answer
32 views

Expressing a solution in interval notation

I am faced with this problem. I am told to express the answer in interval notation. |3x| > 12 I solve like usual, by doing this: ...
0
votes
3answers
44 views

Do you flip the inequality sign if multiplying a quadratic equation by $-1$?

$$(-1)(-x^{ 2 }+3x+18)<0(-1)$$ becomes $$x^{ 2 }-3x-18>0\quad ?$$ I want to confirm before proceeding in solving a quadratic inequality.
2
votes
4answers
65 views

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 ...
3
votes
6answers
457 views

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 ?
3
votes
3answers
100 views

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 ...
1
vote
1answer
56 views

Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says 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 ...
0
votes
1answer
29 views

Arithmetic and Geometric Mean Inequalities [closed]

Can someone help me to understand the logic of: $$\sqrt{ab} \le \frac{a+b}{2}$$ Proof: ?
3
votes
4answers
90 views

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 ...
0
votes
2answers
38 views

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 ...
0
votes
0answers
43 views

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?
2
votes
1answer
84 views

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 ...
0
votes
3answers
62 views

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 ...
8
votes
6answers
575 views

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 ...
1
vote
2answers
41 views

Radical Inequality

$\sqrt{2x-1}$ + $\sqrt{3x-2}$ > $\sqrt{4x-3}$ + $\sqrt{5x-4}$ I have attempted to solve this by squaring each side, resulting in $5x + 2\sqrt{2x-1}\sqrt{3x-2} - 3 > 9x + 2\sqrt{(4x-3)(5x-4)} - 7 ...
2
votes
4answers
839 views

How to solve inequalities with absolute values on both sides?

If you have an inequality that has two absolute value bars like $|4x+1|<|3x|$, how do you go about doing this? I know that if $4x+1<3x$, then those $x$'s will work but what else do I do? I think ...
5
votes
5answers
155 views

How to solve this inequality? From MSU entrance exam '66

$\frac{\log _{10}\left(2\right)}{\log _{10}\left(\sin \left(x\right)\right)}\le \frac{\log _{10}\left(4\sin ^2\left(x\right)\right)}{\log _{10}\left(\sin \left(x\right)\right)}$ From the title. Not ...
1
vote
2answers
49 views

Given $(x+3)(y−4)=0$, what is the relationship between $xy$ and $-12$?

Given $(x+3)(y−4)=0 $ Quantity $A = xy $ Quantity $B = -12 $ A Quantity $A$ is greater. B Quantity $B$ is greater. C The two quantities are equal. D The relationship cannot be determined from ...
0
votes
0answers
30 views

Comparing Fractional Numbers

Does a formula exist for comparing two fractional numbers, without resolving to using anything other than integers and fractions? (Thus not real numbers). In other words: given $\dfrac{a}{b}$ and ...
0
votes
1answer
49 views

$\left | -(x+2)^2+6(x+2) \right |>13$

I did $-(x+2)^2+6(x+2)>13$ and $-(x+2)^2+6(x+2)< -13$. The first inequality had complex solutions and therefore can be disregarded but the second one has two real solutions, $x \approx -3.7$ and ...
0
votes
1answer
32 views

How to prove $|q|\ge 1 \Rightarrow |a|\ge |d|$?

Let $a,d,q \in \mathbb{Z}$ and $a=dq$ How do I show that $|q| \ge 1 \Rightarrow |a| \ge |d|$? I've tried: $|q|\ge 1 \Rightarrow (q>1 \text{, if } q>0) \text { or } (-q>1 \text{, if } ...
1
vote
4answers
128 views

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.
0
votes
1answer
64 views

inequality funny question

I'm not sure what they want here: solve the inequality in realtion to $x$ for various values of $a$ : $\frac{(a+2)x}{a-1} - \frac{2}{3} < 2x-1$
2
votes
2answers
62 views

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 ...
5
votes
1answer
40 views

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 $x,y,z,t>0$?
0
votes
0answers
42 views

$\sum$ of binomial coefficients inequality

Let $m,n$ be positive integers with $m>n$. When is it true that $$m\cdot 5^{m-1}\cdot 3+\binom{m}{3}\cdot 5^{m-3}\cdot 3^3\cdot 2+\cdots +\binom{m}{2k+1}\cdot m^{m-2k-1}\cdot 3^{2k+1}\cdot ...
0
votes
1answer
43 views

Solve $\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$

Solve the inequality $$\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$$ A) $X<1$ B) $X>1$ C) $X>-1$ D) $X<-1$ E) $X>-2$
-1
votes
1answer
54 views

Finding two sided bounds on $(x+y)/(xy)$ given inequalities for $x$ and $y$

Given $\dfrac{1}{6} < x < \dfrac{1}{2}$ and $\dfrac{1}{7} < y < \dfrac{1}{3}$, can we determine bounds for $\dfrac{x+y}{xy}$?
2
votes
2answers
122 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
0
votes
2answers
63 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
10
votes
3answers
67 views

Solve inequality: $-5 < \frac{1}{x} < 0$

Solve inequality: $-5 < \frac{1}{x} < 0$ I thought about how I can solve this. If I multiply all sides by $x$ I'm afraid I'm removing the answer, cause $\frac{x}{x}=1$. And when $x$ 'leaves' ...
3
votes
2answers
81 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 ...
1
vote
4answers
89 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 ...
2
votes
2answers
55 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 ...
2
votes
1answer
44 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 ...
-7
votes
2answers
77 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
129 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
406 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 ...