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Solve: \[\dfrac{{x + 3}}{{x - 2}} - \dfrac{{1 - x}}{x} = \dfrac{{17}}{x}\] \[\left( {x \ne 0,2} \right)\].

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Last updated date: 19th Jul 2024
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Answer
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Hint: Here we need to find the value of the given variable by solving the given equation. For that, we will solve the given equation by first adding the fractions in the left hand side expression. Then we will use different mathematical operations like addition, subtraction, multiplication and division. After simplifying the terms, we will separate the variable and constants. From there, we will get the required value of the given variable.

Complete step by step solution:
Here we need to find the value of the given variable by solving the given equation \[\dfrac{{x + 3}}{{x - 2}} - \dfrac{{1 - x}}{x} = \dfrac{{17}}{x}\].
Now, we will first take LCM on the left hand side expression.
\[ \Rightarrow \dfrac{{\left( {x + 3} \right)x - \left( {x - 2} \right)\left( {1 - x} \right)}}{{\left( {x - 2} \right) \cdot x}} = \dfrac{{17}}{x}\]
Using the distributive property of multiplication to multiply the terms, we get
\[ \Rightarrow \dfrac{{{x^2} + 3x - x \cdot \left( {1 - x} \right) + 2 \cdot \left( {1 - x} \right)}}{{\left( {x - 2} \right) \cdot x}} = \dfrac{{17}}{x}\]
Again using the distributive property of multiplication, we get
\[ \Rightarrow \dfrac{{{x^2} + 3x - x + {x^2} + 2 - 2x}}{{\left( {x - 2} \right) \cdot x}} = \dfrac{{17}}{x}\]
Now, we will add and subtract the like terms together. Therefore, we get
\[ \Rightarrow \dfrac{{2{x^2} + 2}}{{\left( {x - 2} \right) \cdot x}} = \dfrac{{17}}{x}\]
On simplifying the terms, we get
\[ \Rightarrow \dfrac{{2{x^2} + 2}}{{x - 2}} = \dfrac{{17}}{x}\]
Now, we will cross multiply the terms, so we get
\[ \Rightarrow 2{x^2} + 2 = 17\left( {x - 2} \right)\]
Again using the distributive property of multiplication, we get
\[ \Rightarrow 2{x^2} + 2 = 17x - 34\]
On moving all the terms to one side, we get
\[ \Rightarrow 2{x^2} - 17x + 36 = 0\]
The above equation is a quadratic equation, so we will find the value of the variable by using the factorization method.
Now, we will factorize the quadratic equation by splitting the middle terms. Therefore, we get
\[ \Rightarrow 2{x^2} - 8x - 9x + 36 = 0\]
Now, we will take common factors from the first two terms and the last two terms.
\[ \Rightarrow 2x\left( {x - 4} \right) - 9\left( {x - 4} \right) = 0\]
Factoring out the common terms, we get
\[ \Rightarrow \left( {2x - 9} \right)\left( {x - 4} \right) = 0\]
Applying zero product property, we get
\[ \Rightarrow \left( {2x - 9} \right) = 0\] and \[\left( {x - 4} \right) = 0\]
Now, we will solve these equations individually.
First we will solve \[2x - 9 = 0\].
Adding 9 on both the sides
\[\begin{array}{l} \Rightarrow 2x - 9 + 9 = 0 + 9\\ \Rightarrow 2x = 9\end{array}\]
Now, we will divide both sides by 2. So, we gey
\[ \Rightarrow \dfrac{{2x}}{2} = \dfrac{9}{2}\]
\[ \Rightarrow x = \dfrac{9}{2}\]
Now, we will solve \[x - 4 = 0\].
Adding 4 on both the sides, we get
\[\begin{array}{l} \Rightarrow x - 4 + 4 = 0 + 4\\ \Rightarrow x = 4\end{array}\]

Therefore, the values of the variable \[x\] are \[\dfrac{9}{2}\] and 4.
Hence, this is the required solution to the given variable.


Note:
An algebraic equation is defined as equality which consists of the variables and constants and these are combined using different mathematical operations. We have simplified the given algebraic equation to get the quadratic equation. A quadratic equation is defined as an equation that has the highest degree of variable as 2 and also has 2 solutions. Here, in the question, it is given that \[x\] is not equal to 0 or 2 because these values will make the denominator 0 and hence the fraction becomes undefined.