Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Prove that (1)/(sqrt2-1)+(2)/(sqrt3+1)=>sqrt2+sqrt3

Prove that 12−1+23+1⇒2+3 \frac{1}{\sqrt{2}-1}+\frac{2}{\sqrt{3}+1} \Rightarrow \sqrt{2}+\sqrt{3}

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 1right chevron icon
Solve advanced linear inequalitiesright chevron icon

Full solution

Q. Prove that 12−1+23+1⇒2+3 \frac{1}{\sqrt{2}-1}+\frac{2}{\sqrt{3}+1} \Rightarrow \sqrt{2}+\sqrt{3}
  1. Rationalize first fraction: To simplify the left side of the inequality, we will rationalize the denominators of each fraction.\newlineRationalize the first fraction: (1)/(2−1)×(2+1)/(2+1)=(2+1)/(2−1)=2+1(1)/(\sqrt{2}-1) \times (\sqrt{2}+1)/(\sqrt{2}+1) = (\sqrt{2}+1)/(2-1) = \sqrt{2}+1.
  2. Rationalize second fraction: Rationalize the second fraction: (\frac{\(2\)}{\sqrt{\(3\)}+\(1\)}) \cdot (\frac{\sqrt{\(3\)}\(-1\)}{\sqrt{\(3\)}\(-1\)}) = \frac{\(2\)\sqrt{\(3\)}\(-2\)}{\(3\)\(-1\)} = \frac{\(2\)\sqrt{\(3\)}\(-2\)}{\(2\)} = \sqrt{\(3\)}\(-1\.
  3. Combine rationalized fractions: Combine the two rationalized fractions: (2+1)+(3−1)(\sqrt{2}+1) + (\sqrt{3}-1).
  4. Simplify combined expression: Simplify the combined expression: (2+1)+(3−1)=2+3(\sqrt{2}+1) + (\sqrt{3}-1) = \sqrt{2} + \sqrt{3}.
  5. Simplified left side: Now we have the simplified left side of the inequality: 2+3\sqrt{2} + \sqrt{3}. The original inequality to prove was 12−1+23+1≥2+3\frac{1}{\sqrt{2}-1}+\frac{2}{\sqrt{3}+1} \geq \sqrt{2}+\sqrt{3}. After simplification, we have 2+3≥2+3\sqrt{2} + \sqrt{3} \geq \sqrt{2} + \sqrt{3}.
  6. Prove inequality: Since both sides of the inequality are the same, the inequality holds true.

More problems from Solve advanced linear inequalities

Question
−2x+by=59 -2 x+b y=59 \newline5x−y=1 5 x-y=1 \newlineIf the system of equations has exactly one solution (x,y) (x, y) and x=2 x=2 , what is the value of b b ?
Get tutor helpright-arrow

Posted 2 months ago

Question
Evaluate.\newline23−1283= \frac{\sqrt[3]{2}}{\sqrt[3]{-128}}=
Get tutor helpright-arrow

Posted 6 days ago

Question
Evaluate.\newline241624= \frac{\sqrt[4]{2}}{\sqrt[4]{162}}=
Get tutor helpright-arrow

Posted 3 months ago

Question
Evaluate.\newline−165⋅215= \sqrt[5]{-16} \cdot 2^{\frac{1}{5}}=
Get tutor helpright-arrow

Posted 2 months ago

Question
Solve for y y .\newliney−13y+7=110y=□ \begin{array}{l} \frac{y-1}{3 y+7}=\frac{1}{10} \\ y=\square \end{array}
Get tutor helpright-arrow

Posted 3 months ago

Question
Solve for x x .\newlineYour answer must be simplified.\newline−30<x−4 -30<\frac{x}{-4}
Get tutor helpright-arrow

Posted 2 months ago

Question
Solve for x x .\newlineYour answer must be simplified.\newline−36+x<8 -36+x<8
Get tutor helpright-arrow

Posted 2 months ago

Question
Solve for x x .\newlineYour answer must be simplified.\newline2x<15 2 x<15
Get tutor helpright-arrow

Posted 3 months ago

Question
Solve for x x .\newlineYour answer must be simplified.\newlinex−6≤3 x-6 \leq 3
Get tutor helpright-arrow

Posted 1 month ago

Question
Solve for x x .\newlineYour answer must be simplified.\newline7>x4 7>\frac{x}{4}
Get tutor helpright-arrow

Posted 3 months ago

View all (118)