Sure, let's solve the equation step-by-step.
Given:
[tex]\[ 3(2x - 9) - 4(3x + 4) = 2(x - 12) + \frac{3}{7} \][/tex]
1. Expand the Expressions:
- Distribute the constants within the parentheses.
For [tex]\( 3(2x - 9) \)[/tex]:
[tex]\[ 3 \cdot 2x - 3 \cdot 9 = 6x - 27 \][/tex]
For [tex]\( -4(3x + 4) \)[/tex]:
[tex]\[ -4 \cdot 3x - 4 \cdot 4 = -12x - 16 \][/tex]
For [tex]\( 2(x - 12) \)[/tex]:
[tex]\[ 2 \cdot x - 2 \cdot 12 = 2x - 24 \][/tex]
Substitute these expanded forms back into the equation:
[tex]\[ 6x - 27 - 12x - 16 = 2x - 24 + \frac{3}{7} \][/tex]
2. Combine Like Terms:
- On the left side: [tex]\( 6x - 12x \)[/tex] and [tex]\(-27 - 16\)[/tex]:
[tex]\[ 6x - 12x = -6x \][/tex]
[tex]\[ -27 - 16 = -43 \][/tex]
So, the left side simplifies to:
[tex]\[ -6x - 43 \][/tex]
Now the equation looks like:
[tex]\[ -6x - 43 = 2x - 24 + \frac{3}{7} \][/tex]
3. Get Rid of the Fraction:
- To eliminate the fraction, multiply every term by 7:
[tex]\[ 7(-6x - 43) = 7(2x - 24) + 3 \][/tex]
Expanding both sides:
[tex]\[ -42x - 301 = 14x - 168 + 3 \][/tex]
4. Simplify Both Sides:
Combine like terms on the right side:
[tex]\[ 14x - 168 + 3 = 14x - 165 \][/tex]
Now the equation is:
[tex]\[ -42x - 301 = 14x - 165 \][/tex]
5. Combine Like Terms to Isolate [tex]\( x \)[/tex]:
- Move all [tex]\( x \)[/tex]-terms to one side and constants to the other.
Adding [tex]\( 42x \)[/tex] to both sides:
[tex]\[ -301 = 14x + 42x - 165 \][/tex]
[tex]\[ -301 = 56x - 165 \][/tex]
Adding 165 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ -301 + 165 = 56x \][/tex]
[tex]\[ -136 = 56x \][/tex]
6. Solve for [tex]\( x \)[/tex]:
Divide both sides by 56:
[tex]\[ x = \frac{-136}{56} \][/tex]
Simplify the fraction:
[tex]\[ x = -\frac{136}{56} = -\frac{34}{14} = -\frac{17}{7} \approx -2.42857142857143 \][/tex]
So, the solution to the equation [tex]\(3(2 x-9) - 4(3 x+4) = 2(x-12) + \frac{3}{7}\)[/tex] is:
[tex]\[ x \approx -2.42857142857143 \][/tex]
