To solve the equation [tex]\( 2(5 + 2x) - 25 - x = -8(2x - 1) + 32 + 8x \)[/tex], let's go through it step by step.
This is our given equation:
[tex]\[ 2(5 + 2x) - 25 - x = -8(2x - 1) + 32 + 8x \][/tex]
### Step 1: Expand both sides of the equation
First, expand the expressions inside the parentheses.
For the left side:
[tex]\[ 2(5 + 2x) = 2 \cdot 5 + 2 \cdot 2x = 10 + 4x \][/tex]
Substitute this back into the equation:
[tex]\[ 10 + 4x - 25 - x \][/tex]
Simplify the left side:
[tex]\[ 10 + 4x - 25 - x = -15 + 3x \][/tex]
Now for the right side:
[tex]\[ -8(2x - 1) + 32 + 8x \][/tex]
Expand the term:
[tex]\[ -8(2x - 1) = -8 \cdot 2x + (-8) \cdot (-1) = -16x + 8 \][/tex]
Substitute this back into the equation:
[tex]\[ -16x + 8 + 32 + 8x \][/tex]
Combine like terms on the right side:
[tex]\[ -16x + 8 + 32 + 8x = -16x + 8x + 40 = -8x + 40 \][/tex]
### Step 2: Set both expressions equal to each other
Now we have:
[tex]\[ -15 + 3x = -8x + 40 \][/tex]
### Step 3: Combine like terms to solve for [tex]\( x \)[/tex]
First, move all terms involving [tex]\( x \)[/tex] to one side and constants to the other:
Add [tex]\( 8x \)[/tex] to both sides:
[tex]\[ -15 + 3x + 8x = 40 \][/tex]
This simplifies to:
[tex]\[ -15 + 11x = 40 \][/tex]
Next, add 15 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 11x = 40 + 15 \][/tex]
[tex]\[ 11x = 55 \][/tex]
Finally, divide by 11 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{55}{11} \][/tex]
[tex]\[ x = 5 \][/tex]
### Conclusion:
So, the solution to the equation [tex]\( 2(5 + 2x) - 25 - x = -8(2x - 1) + 32 + 8x \)[/tex] is
[tex]\[ x = 5 \][/tex]
