r) $\frac{2 x-1}{2 x+1}-\frac{2 x+1}{2 x-1}=-2 \frac{2}{3}$
Answer (2)
To solve the equation 2 x + 1 2 x − 1 − 2 x − 1 2 x + 1 = − 2 3 2 , we converted the mixed number to an improper fraction and combined fractions. This led to the quadratic equation 4 x 2 − 3 x − 1 = 0 , which we solved to find x = − 4 1 and x = 1 as valid solutions.
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Rewrite the mixed number as an improper fraction and combine the fractions on the left side.
Simplify the equation to a quadratic equation: 4 x 2 − 3 x − 1 = 0 .
Solve the quadratic equation by factoring: ( 4 x + 1 ) ( x − 1 ) = 0 .
The solutions are x = − 4 1 and x = 1 , which satisfy the initial restrictions. Thus, the solutions are − 4 1 , 1 .
Explanation
Rewrite the mixed number We are given the equation 2 x + 1 2 x − 1 − 2 x − 1 2 x + 1 = − 2 3 2 Our goal is to solve for x . First, we need to rewrite the mixed number as an improper fraction: − 2 3 2 = − 3 8 .
Combine fractions and simplify Now, we find a common denominator and combine the fractions on the left side of the equation: ( 2 x + 1 ) ( 2 x − 1 ) ( 2 x − 1 ) 2 − ( 2 x + 1 ) 2 = − 3 8 Expanding the squares in the numerator, we get: 4 x 2 − 1 ( 4 x 2 − 4 x + 1 ) − ( 4 x 2 + 4 x + 1 ) = − 3 8 Simplifying the numerator: 4 x 2 − 1 − 8 x = − 3 8 Dividing both sides by -8: 4 x 2 − 1 x = 3 1 Cross-multiplying: 3 x = 4 x 2 − 1 Rearranging the equation into a quadratic equation: 4 x 2 − 3 x − 1 = 0
Solve the quadratic equation Now, we solve the quadratic equation 4 x 2 − 3 x − 1 = 0 for x . We can use the quadratic formula or factoring. Let's try factoring: ( 4 x + 1 ) ( x − 1 ) = 0 So, the possible solutions are: 4 x + 1 = 0 ⇒ x = − 4 1 x − 1 = 0 ⇒ x = 1 We need to check if these solutions are valid, considering the restrictions x = ± 2 1 . Both solutions, x = − 4 1 and x = 1 , satisfy this condition.
Final Answer Therefore, the solutions to the equation are x = − 4 1 and x = 1 .
Examples
When solving problems involving rates and proportions, such as comparing the efficiency of two machines working together, you might encounter equations similar to the one we solved. For instance, if two machines complete a task at different rates, the combined rate can be modeled with rational expressions. Solving such equations helps determine the time it takes for them to complete the task together or to optimize their individual contributions for maximum efficiency. This type of problem arises in various fields, including engineering, logistics, and resource management, where understanding and optimizing rates and proportions are crucial for efficient operations.
