To solve the equation [tex]\(-7 - 7(-x - 1) = 6 - (-2x - 3)\)[/tex], we'll go through a detailed, step-by-step process.
Step 1: Distribute the multiplications.
- On the left side of the equation, distribute [tex]\(-7\)[/tex] through [tex]\(-x - 1\)[/tex]:
[tex]\[
-7 - 7(-x - 1) = -7 - 7(-x) - 7(1) = -7 + 7x - 7
\][/tex]
Simplify this expression:
[tex]\[
-7 + 7x - 7 = 7x - 14
\][/tex]
- On the right side of the equation, distribute [tex]\(-1\)[/tex] through [tex]\(-2x - 3\)[/tex]:
[tex]\[
6 - (-2x - 3) = 6 + 2x + 3
\][/tex]
Simplify this expression:
[tex]\[
6 + 2x + 3 = 9 + 2x
\][/tex]
So now our equation is:
[tex]\[
7x - 14 = 9 + 2x
\][/tex]
Step 2: Isolate the variable term.
- First, move the term containing [tex]\(x\)[/tex] on the right side to the left side by subtracting [tex]\(2x\)[/tex] from both sides:
[tex]\[
7x - 2x - 14 = 9 + 2x - 2x
\][/tex]
Simplify this equation:
[tex]\[
5x - 14 = 9
\][/tex]
Step 3: Solve for [tex]\(x\)[/tex].
- Next, move the constant term [tex]\(-14\)[/tex] on the left side to the right side by adding [tex]\(14\)[/tex] to both sides:
[tex]\[
5x - 14 + 14 = 9 + 14
\][/tex]
Simplify this equation:
[tex]\[
5x = 23
\][/tex]
- Finally, solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(5\)[/tex]:
[tex]\[
x = \frac{23}{5}
\][/tex]
Simplify the fraction:
[tex]\[
x = 4.6
\][/tex]
Thus, the solution to the equation is:
[tex]\[
x = 4.6
\][/tex]
