- #1
Albert1
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$(0<x<3), \sqrt {1+x^2}+\sqrt {9+(3-x)^2} =5$
find :$x$
find :$x$
What is the value of x for the inequality problem?
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In summary, an inequality problem involves finding the value of a variable that satisfies a condition using greater than or less than symbols. The purpose of finding x in an inequality problem is to determine the range of values that the variable can take to make the inequality statement true. To solve an inequality problem, the same steps as solving an equation are followed, with an additional step of checking if the value found satisfies the inequality. There can be more than one solution for an inequality problem, and common mistakes to avoid include not following the correct order of operations and not checking the solution for satisfaction of the given inequality.
- #2
anemone
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Albert said:
If we let $x=\tan y$ and notice that $1+(\tan y)^2=\sec^2 y$, we get:
$\sqrt {1+(\tan y)^2}+\sqrt {9+(3-\tan y)^2} =5$
$\sec y+\sqrt {9+(3-\tan y)^2} =5$
$\sqrt {9+(3-\tan y)^2} =5-\sec y$
Square both sides of the equation and simplify, we have:
$-3\tan y =4-5\sec y$
Squaring again and use the identity $1+(\tan y)^2=\sec^2 y$ we obtain:
$(4\sec y-5)^2=0$
In other words, $\sec y =\dfrac{5}{4}$ and this implies $\tan y= \dfrac{3}{4}$ or $x=\dfrac{3}{4}$.
$\sqrt {1+(\tan y)^2}+\sqrt {9+(3-\tan y)^2} =5$
$\sec y+\sqrt {9+(3-\tan y)^2} =5$
$\sqrt {9+(3-\tan y)^2} =5-\sec y$
Square both sides of the equation and simplify, we have:
$-3\tan y =4-5\sec y$
Squaring again and use the identity $1+(\tan y)^2=\sec^2 y$ we obtain:
$(4\sec y-5)^2=0$
In other words, $\sec y =\dfrac{5}{4}$ and this implies $\tan y= \dfrac{3}{4}$ or $x=\dfrac{3}{4}$.
- #3
MarkFL
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Here is my solution:
Arrange the equation as follows:
\(\displaystyle \sqrt{9+(3-x)^2}=5-\sqrt{1+x^2}\)
Both sides are positive, so squaring yields:
\(\displaystyle 9+(3-x)^2=25-10\sqrt{1+x^2}+1+x^2\)
Expand squared binomial and collect like terms:
\(\displaystyle 4+3x=5\sqrt{1+x^2}\)
Square again:
\(\displaystyle 16+24x+9x^2=25\left(1+x^2 \right)\)
\(\displaystyle 16x^2-24x+9=0\)
\(\displaystyle (4x-3)^2=0\)
\(\displaystyle x=\frac{3}{4}\)
\(\displaystyle \sqrt{9+(3-x)^2}=5-\sqrt{1+x^2}\)
Both sides are positive, so squaring yields:
\(\displaystyle 9+(3-x)^2=25-10\sqrt{1+x^2}+1+x^2\)
Expand squared binomial and collect like terms:
\(\displaystyle 4+3x=5\sqrt{1+x^2}\)
Square again:
\(\displaystyle 16+24x+9x^2=25\left(1+x^2 \right)\)
\(\displaystyle 16x^2-24x+9=0\)
\(\displaystyle (4x-3)^2=0\)
\(\displaystyle x=\frac{3}{4}\)
- #4
Albert1
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Albert said:
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CellCoree
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To solve this inequality problem, we can start by simplifying the equation. First, we can expand the square roots to get $\sqrt{1+x^2} = \sqrt{x^2+1}$ and $\sqrt{9+(3-x)^2} = \sqrt{x^2-6x+18}$. This gives us the equation $\sqrt{x^2+1}+\sqrt{x^2-6x+18}=5$.
Next, we can square both sides of the equation to eliminate the square roots. This gives us $x^2+1+x^2-6x+18+2\sqrt{(x^2+1)(x^2-6x+18)}=25$. Simplifying further, we get $2x^2-6x+19+2\sqrt{x^4-6x^3+19x^2-108x+18} = 25$.
We can then move all terms to one side of the equation and simplify to get $2x^2-6x-6=2\sqrt{x^4-6x^3+19x^2-108x+18}$. We can again square both sides to eliminate the square root, giving us $4x^4-24x^3+124x^2-696x+108=0$.
Finally, we can factor this equation to get $(2x-3)(2x^3-18x^2+92x-36)=0$. Solving for $x$, we get $x=\frac{3}{2}$ or $x\approx2.7$. However, since the given inequality states that $x$ must be less than 3, the only valid solution is $x=\frac{3}{2}$.
FAQ: What is the value of x for the inequality problem?
What is an inequality problem?
An inequality problem is a mathematical problem that involves finding the value of a variable that satisfies a condition involving greater than or less than symbols.
What is the purpose of finding x in an inequality problem?
The purpose of finding x in an inequality problem is to determine the range of values that the variable can take in order to make the given inequality statement true.
How do I solve an inequality problem?
To solve an inequality problem, you need to follow the same steps as you would in solving an equation, but with one additional step. After solving for x, you need to check if the value you found satisfies the given inequality. If it does, then that is the solution. If not, then you need to find another value of x that satisfies the inequality.
Can there be more than one solution for an inequality problem?
Yes, there can be more than one solution for an inequality problem. In fact, there can be infinitely many solutions, depending on the given inequality.
What are the common mistakes to avoid when solving an inequality problem?
The common mistakes to avoid when solving an inequality problem include not following the correct order of operations, forgetting to reverse the inequality symbol when multiplying or dividing by a negative number, and not checking the solution to see if it satisfies the given inequality.