Certainly, let's solve each of these equations step-by-step.
### 1. Solve [tex]\(2(x-4)=3(x+2)\)[/tex]
Equation: [tex]\(2(x-4)=3(x+2)\)[/tex]
Step-by-Step:
1. Distribute the constants within the brackets:
[tex]\[
2x - 8 = 3x + 6
\][/tex]
2. Isolate the terms involving [tex]\( x \)[/tex] on one side:
[tex]\[
2x - 3x = 6 + 8
\][/tex]
3. Simplify the equation:
[tex]\[
-x = 14
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -14
\][/tex]
### 2. Solve [tex]\(-3(x-1)=2(x-4)\)[/tex]
Equation: [tex]\(-3(x-1)=2(x-4)\)[/tex]
Step-by-Step:
1. Distribute the constants within the brackets:
[tex]\[
-3x + 3 = 2x - 8
\][/tex]
2. Isolate the terms involving [tex]\( x \)[/tex] on one side:
[tex]\[
-3x - 2x = -8 - 3
\][/tex]
3. Simplify the equation:
[tex]\[
-5x = -11
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{11}{5}
\][/tex]
### 3. Solve [tex]\(-3(x+2)-3x=3(x+1)\)[/tex]
Equation: [tex]\(-3(x+2)-3x=3(x+1)\)[/tex]
Step-by-Step:
1. Distribute the constants within the brackets:
[tex]\[
-3x - 6 - 3x = 3x + 3
\][/tex]
2. Combine the [tex]\( x \)[/tex] terms on the left-hand side:
[tex]\[
-6x - 6 = 3x + 3
\][/tex]
3. Isolate the terms involving [tex]\( x \)[/tex] on one side:
[tex]\[
-6x - 3x = 3 + 6
\][/tex]
4. Simplify the equation:
[tex]\[
-9x = 9
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -1
\][/tex]
### 4. Solve [tex]\(-3(x+2)=3(x+1)\)[/tex]
Equation: [tex]\(-3(x+2)=3(x+1)\)[/tex]
Step-by-Step:
1. Distribute the constants within the brackets:
[tex]\[
-3x - 6 = 3x + 3
\][/tex]
2. Isolate the terms involving [tex]\( x \)[/tex] on one side:
[tex]\[
-3x - 3x = 3 + 6
\][/tex]
3. Simplify the equation:
[tex]\[
-6x = 9
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -\frac{3}{2}
\][/tex]
### Summary of Solutions
- For the first equation [tex]\(2(x-4)=3(x+2)\)[/tex], [tex]\(x = -14\)[/tex].
- For the second equation [tex]\(-3(x-1)=2(x-4)\)[/tex], [tex]\(x = \frac{11}{5}\)[/tex].
- For the third equation [tex]\(-3(x+2)-3x=3(x+1)\)[/tex], [tex]\(x = -1\)[/tex].
- For the fourth equation [tex]\(-3(x+2)=3(x+1)\)[/tex], [tex]\(x = -\frac{3}{2}\)[/tex].
