Question

solve for ( x ). if there are multiple solutions, enter them as a list of values separated by commas. if there are no solutions, enter none. (\frac{4}{5x + 3} = \frac{-1}{-7x - 9}) ( x = ) check answer

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{4}{5x + 3}=\frac{-1}{-7x-9}\), we can use the cross - multiplication property of fractions. If \(\frac{a}{b}=\frac{c}{d}\) (where \(b
eq0\) and \(d
eq0\)), then \(a\times d = b\times c\).
Applying this property to our equation, we get \(4\times(-7x - 9)=-1\times(5x + 3)\).

Step2: Expand both sides of the equation

First, expand the left - hand side: \(4\times(-7x)-4\times9=-28x-36\).
Then, expand the right - hand side: \(-1\times5x+(-1)\times3=-5x - 3\).
So our equation becomes \(-28x-36=-5x - 3\).

Step3: Move the \(x\) terms to one side and constants to the other side

Add \(28x\) to both sides of the equation: \(-28x+28x-36=-5x + 28x-3\), which simplifies to \(-36 = 23x-3\).
Then, add 3 to both sides: \(-36 + 3=23x-3 + 3\), so \(-33 = 23x\).

Step4: Solve for \(x\)

Divide both sides of the equation by 23: \(x=\frac{-33}{23}\).
We need to check if this solution makes the denominators of the original fractions zero.
For the denominator \(5x + 3\): \(5\times(\frac{-33}{23})+3=\frac{-165}{23}+\frac{69}{23}=\frac{-165 + 69}{23}=\frac{-96}{23}
eq0\).
For the denominator \(-7x-9\): \(-7\times(\frac{-33}{23})-9=\frac{231}{23}-\frac{207}{23}=\frac{231 - 207}{23}=\frac{24}{23}
eq0\).
So \(x = \frac{-33}{23}\) is a valid solution.

Answer:

\(-\frac{33}{23}\)