Sure! Let's solve the equation step-by-step.
We are given the equation:
[tex]\[
\frac{4x - 5}{x + 2} = \frac{8x - 1}{2x + 1}
\][/tex]
Step 1: Cross-multiply to eliminate the fractions
[tex]\[
(4x - 5)(2x + 1) = (8x - 1)(x + 2)
\][/tex]
Step 2: Expand both sides of the equation
Let's expand the left-hand side:
[tex]\[
(4x - 5)(2x + 1) = 4x \cdot 2x + 4x \cdot 1 - 5 \cdot 2x - 5 \cdot 1
\][/tex]
[tex]\[
= 8x^2 + 4x - 10x - 5
\][/tex]
[tex]\[
= 8x^2 - 6x - 5
\][/tex]
Now, let's expand the right-hand side:
[tex]\[
(8x - 1)(x + 2) = 8x \cdot x + 8x \cdot 2 - 1 \cdot x - 1 \cdot 2
\][/tex]
[tex]\[
= 8x^2 + 16x - x - 2
\][/tex]
[tex]\[
= 8x^2 + 15x - 2
\][/tex]
Step 3: Set the two expanded equations equal to each other
[tex]\[
8x^2 - 6x - 5 = 8x^2 + 15x - 2
\][/tex]
Step 4: Subtract [tex]\(8x^2\)[/tex] from both sides of the equation
[tex]\[
-6x - 5 = 15x - 2
\][/tex]
Step 5: Move all the [tex]\(x\)[/tex]-terms to one side and the constants to the other side
[tex]\[
-6x - 15x = -2 + 5
\][/tex]
[tex]\[
-21x = 3
\][/tex]
Step 6: Solve for [tex]\(x\)[/tex]
[tex]\[
x = \frac{3}{-21}
\][/tex]
[tex]\[
x = -\frac{1}{7}
\][/tex]
So, the solution to the equation is:
[tex]\[
x = -\frac{1}{7}
\][/tex]
