Question

Can someone teach me how to solve rational equations step-by-step. This is not a graded assignment.

Answer 1

To solve this question, follow the steps below.

Step 01: Factor the denominator of each fraction using (b - 3).

First,

$\begin{gathered} b^2-3b \\ b(b-3) \end{gathered}$

Second,

$\begin{gathered} b^3-3b^2 \\ b^2(b-3) \end{gathered}$

Third,

$(b-3)$

So, the expression is:

$\frac{1}{b(b-3)}+\frac{b^2-3b-4}{b^2(b-3)}=\frac{1}{(b-3)}$

Step 02: Write all fractions together using b²(b-3) as a denominator.

$\frac{b*1+b^2-3b-4=b^2*1}{b^2(b-3)}$

Solve the multiplications and then the additions/subtractions in the numerator.

$\begin{gathered} \frac{b+b^2-3b-4=b^2}{b^2(b-3)} \\ \frac{b^2-2b-4=b^2}{b^2(b-3)} \end{gathered}$

Step 03: Remove the denominator.

Since both sides are divided by the same number, you can remove it.

$b^2-2b-4=b^2$

Step 04: Write all numbers on the same side of the equation.

To do it, subtract b² from both sides.

$\begin{gathered} b^2-2b-4=b^2 \\ b^2-2b-4-b^2=b^2-b^2 \\ -2b-4=0 \end{gathered}$

Step 05: Solve for b.

To do it, add 4 to both sides and then divide the sides by -2.

$\begin{gathered} -2b-4+4=0+4 \\ -2b=4 \\ \frac{-2b}{-2}=\frac{4}{-2} \\ b=-2 \end{gathered}$

Answer: b = -2.