Sure, let's solve the equation step-by-step:
The given equation is:
[tex]\[
\frac{3}{4}(x-1) - \frac{5}{3}(x-4) = \frac{8(x-6)}{5} \left( x - \frac{5}{12} \right)
\][/tex]
### Step 1: Simplify both sides
Let's begin by expanding and simplifying both sides of the equation.
#### Left-hand side (LHS):
[tex]\[
\frac{3}{4}(x-1) - \frac{5}{3}(x-4)
\][/tex]
First, distribute the coefficients inside the parentheses:
[tex]\[
\frac{3}{4}x - \frac{3}{4} - \frac{5}{3}x + \frac{20}{3}
\][/tex]
Combine like terms:
[tex]\[
\left( \frac{3}{4}x - \frac{5}{3}x \right) + \left( \frac{20}{3} - \frac{3}{4} \right)
\][/tex]
To combine the fractions, we need a common denominator. For [tex]\( x \)[/tex]-terms, the common denominator between 4 and 3 is 12:
[tex]\[
\frac{3}{4}x = \frac{9}{12}x \quad \text{and} \quad \frac{5}{3}x = \frac{20}{12}x
\][/tex]
Combine:
[tex]\[
\frac{9}{12}x - \frac{20}{12}x = \frac{-11}{12}x
\][/tex]
Now for the constants, the common denominator between 3 and 4 is also 12:
[tex]\[
\frac{20}{3} = \frac{80}{12} \quad \text{and} \quad \frac{3}{4} = \frac{9}{12}
\][/tex]
Combine:
[tex]\[
\frac{80}{12} - \frac{9}{12} = \frac{71}{12}
\][/tex]
Putting it all together for LHS:
[tex]\[
\frac{-11}{12}x + \frac{71}{12}
\][/tex]
#### Right-hand side (RHS):
[tex]\[
\frac{8(x-6)}{5} \left( x - \frac{5}{12} \right)
\][/tex]
First, distribute [tex]\( \frac{8}{5}(x-6) \)[/tex] to each term inside the parentheses:
[tex]\[
\frac{8(x-6)}{5} \cdot \left(x\right) - \frac{8(x-6)}{5} \cdot \frac{5}{12}
\][/tex]
Distribute [tex]\( \frac{8}{5} \)[/tex] inside [tex]\( (x-6) \)[/tex]:
[tex]\[
\frac{8x - 48}{5} \cdot x - \frac{8x - 48}{5} \cdot \frac{5}{12}
\][/tex]
Simplify each term:
[tex]\[
\left(\frac{8x^2 - 48x}{5}\right) - \left(\frac{8x - 48}{12}\right)
\][/tex]
Rewrite both terms with common denominator and combine like terms if possible. Let's solve simplification computationally for accurate results.
After simplifying both sides, we set them equal and solve for [tex]\( x \)[/tex].
### Step 2: Solve the Equation
After solving, we find that the values of [tex]\( x \)[/tex] that satisfy the equation are:
[tex]\[
x \approx -0.198264449173973 \quad \text{and} \quad x \approx 6.04201444917397
\][/tex]
Therefore, the solutions to the given equation are:
[tex]\[
x \approx -0.198 \quad \text{and} \quad x \approx 6.042
\][/tex]
These solutions can be verified by substituting them back into the original equation to check for equality on both sides.
