91Ó°ÊÓ

The first step in handling equations with fractions is often to eliminate the denominators. Why do we do this? By removing denominators, you transform a rational equation into a simpler, often linear equation, which is easier to solve.

When faced with a rational equation like \(\frac{2x-7}{x+4}=\frac{5}{x+4}-2\), observe if the fractions share a common denominator. If they do, you can multiply each term by that common denominator to eliminate the fractional parts altogether. This is what we did in the exercise, multiplying every term by \(x+4\) leading to the simpler equation \(2x - 7 = 5 - 2(x + 4)\).

Remember that when there are multiple terms with denominators, you need to apply the least common denominator (LCD) for all of them, which may be a multiple of individual denominators.

Once you've eliminated the denominators, the next step is to simplify the equation. This involves expanding any parentheses and combining like terms. You simplify in order to bring the equation into a form where the variable you want to solve for, \(x\), is easier to isolate.

For the given exercise, after eliminating the denominators, expanding the parentheses (\(5 - 2(x + 4)\)) gave us \(2x - 7 = -2x - 3\). Then we combined like terms by adding \(2x\) to both sides, resulting in a more manageable equation, \(4x - 7 = -3\). Simplification is critical because it lays the groundwork for isolating the variable and finding the solution to the equation.

The final and crucial step is to check your solutions. This step ensures that the solution you've determined actually satisfies the original equation, especially since in rational equations, there can be false solutions introduced by the steps of solving.

To check the solution \(x = 1\), we substitute it back into the original equation and simplify: \(\frac{2(1)-7}{1+4}=\frac{5}{1+4}-2\) simplifies to \(\frac{-5}{5}=\frac{5}{5}-2\), and further to \( -1 = 1 - 2\). Since both sides of the equation are equal after substitution, our solution is correct. Checking is important to avoid accepting 'extraneous solutions' that don't actually satisfy the equation.