Question

Solve each equation, if possible. $$ \frac{-4}{2 x+3}+\frac{1}{x-1}=\frac{1}{(2 x+3)(x-1)} $$

Short answer

x = 3

Step-by-step solution

Identify the common denominator

The given equation is \[ \frac{-4}{2x + 3} + \frac{1}{x - 1} = \frac{1}{(2x + 3)(x - 1)} \]Identify the common denominator for the fractions on the left side. The common denominator is \[ (2x + 3)(x - 1) \]

Rewrite each term with the common denominator

Rewrite the left side of the equation with the common denominator:\[ \frac{-4(x - 1) + (2x + 3)}{(2x + 3)(x - 1)} \]This simplifies to:\[ \frac{-4x + 4 + 2x + 3}{(2x + 3)(x - 1)} \]

Combine the terms in the numerator

Combine the terms of the numerator:\[ \frac{-2x + 7}{(2x + 3)(x - 1)} \]So the simplified equation is:\[ \frac{-2x + 7}{(2x + 3)(x - 1)} = \frac{1}{(2x + 3)(x - 1)} \]

Set the numerators equal

Since the denominators are the same, set the numerators equal to each other:\[ -2x + 7 = 1 \]

Solve the equation for x

Subtract 7 from both sides:\[ -2x + 7 - 7 = 1 - 7 \]This simplifies to:\[ -2x = -6 \]Divide both sides by -2:\[ x = 3 \]

Verify the solution

Substitute \( x = 3 \) back into the original equation to verify it works:\[ \frac{-4}{2(3) + 3} + \frac{1}{3 - 1} = \frac{1}{(2(3) + 3)(3 - 1)} \]Simplify each term:\[ \frac{-4}{9} + \frac{1}{2} = \frac{1}{18} \]\[ \frac{-4}{9} + \frac{9}{18} = \frac{-8 + 9}{18} = \frac{1}{18} \]The left side equals the right side, so \( x = 3 \) is a valid solution.

Key concepts

solving rational equations

Solving rational equations involves equations with fractions that have polynomials in their numerators and denominators. The goal is to find values of the variable that make the equation true.
First, it's crucial to identify the common denominators of all the fractions involved. Then, rewrite each term with this common denominator to simplify the equation.
This process turns the rational equation into a simpler form, often a polynomial equation, that is easier to solve.
Finally, you solve for the variable and check if the solution works by substituting the value back into the original equation.

common denominators

Finding a common denominator is essential when working with rational equations. A common denominator allows us to combine fractions, making the equation easier to solve.
For instance, in the given problem, the denominators are \( 2x + 3 \) and \( x - 1 \). The least common denominator (LCD) is their product: \[ (2x + 3)(x - 1) \]
By expressing all fractions with this common denominator, it becomes possible to combine them. This step ensures uniformity and simplifies the process of solving the equation.

simplifying expressions

Simplifying expressions within rational equations is crucial to make them easier to solve. Here’s how you do it:
Once you've identified the common denominator and rewritten all terms accordingly, simplify the numerators by combining like terms.
Take the equation \[ \frac{-4(x - 1) + (2x + 3)}{(2x + 3)(x - 1)} \] as an example. Simplifying the numerator gives us: \[ -4x + 4 + 2x + 3 = -2x + 7 \]
This step transforms a complicated fraction into a simpler, solvable equation.

verifying solutions

Verifying solutions is a critical part of solving rational equations. Once you find a potential solution, substitute it back into the original equation to ensure it satisfies the equation.
For the solution \( x = 3 \), verify it by substituting back into the original equation: \[ \frac{-4}{2(3) + 3} + \frac{1}{3 - 1} = \frac{1}{(2(3) + 3)(3 - 1)} \]
This simplifies to:
\[ \frac{-4}{9} + \frac{1}{2} = \frac{1}{18} \]
\[ \frac{-4}{9} + \frac{9}{18} = \frac{-8 + 9}{18} = \frac{1}{18} \]
Since both sides of the equation balance, \( x = 3 \) is confirmed as a correct solution. This step ensures accuracy and completeness in solving the problem.