Prove: Inequality $9\gt \sqrt{a-1}+\sqrt{19-3a}+\sqrt{2a+9}$

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In summary, this conversation discusses an inequality that states that 9 is greater than the sum of the square roots of three different expressions. It can be proven using algebraic manipulation and there are restrictions on the value of a for it to hold. This type of inequality has real-life applications in fields such as economics, physics, and engineering. An example is given to illustrate the inequality.
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Prove that $9\gt \sqrt{a-1}+\sqrt{19-3a}+\sqrt{2a+9}$ for all real $a$.
 
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The inequality only makes sense when $1\le a\le 19/3$. By concavity of the square root function on $(0,\infty)$,

$$\sqrt{a-1}+\sqrt{19-3a}+\sqrt{2a+9} < 3\sqrt{\frac{(a-1)+(19-3a)+(2a+9)}{3}} = 3\sqrt{\frac{27}{3}} = 3\cdot 3 = 9.$$
 
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Euge said:
The inequality only makes sense when $1\le a\le 19/3$. By concavity of the square root function on $(0,\infty)$,

$$\sqrt{a-1}+\sqrt{19-3a}+\sqrt{2a+9} < 3\sqrt{\frac{(a-1)+(19-3a)+(2a+9)}{3}} = 3\sqrt{\frac{27}{3}} = 3\cdot 3 = 9.$$

Thanks Euge for your elegant method of proving and thanks too for participating. :cool:
 

FAQ: Prove: Inequality $9\gt \sqrt{a-1}+\sqrt{19-3a}+\sqrt{2a+9}$

What does this inequality mean?

This inequality means that the number 9 is greater than the sum of the square roots of three different expressions.

How can I prove this inequality?

To prove this inequality, you can use algebraic manipulation and properties of square roots to simplify the expression and show that the left side is greater than the right side.

Can you provide an example to illustrate this inequality?

Yes, for example, if we let a = 5, then the inequality becomes 9 > √4 + √4 + √19, which simplifies to 9 > 2 + 2 + √19, or 9 > 4 + √19, which is true since 9 is greater than 4 + √19 which is approximately 7.3589.

Are there any restrictions on the value of a for this inequality to hold?

Yes, there are restrictions on the value of a. The expression inside the square roots must be non-negative, so a-1, 19-3a, and 2a+9 must all be greater than or equal to 0. This means that a must be greater than or equal to 1, and less than or equal to 19/3.

What real-life applications does this inequality have?

This type of inequality can be used in various fields, such as economics, physics, and engineering, to represent relationships between different variables. For example, it can be used to model production costs, physical constraints, or optimization problems.

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