Sure, let's solve each of these equations step-by-step.
### Equation 1: [tex]\( 7x - 4 = -3x + 6 \)[/tex]
1. Combine like terms by adding [tex]\(3x\)[/tex] to both sides:
[tex]\[ 7x - 4 + 3x = 6 \][/tex]
[tex]\[ 10x - 4 = 6 \][/tex]
2. Add 4 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 10x - 4 + 4 = 6 + 4 \][/tex]
[tex]\[ 10x = 10 \][/tex]
3. Divide both sides by 10 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{10}{10} \][/tex]
[tex]\[ x = 1 \][/tex]
So, the solution is [tex]\( x = 1 \)[/tex].
### Equation 2: [tex]\( 3(x + 2) = 2(x + 1) \)[/tex]
1. Distribute the constants inside the parentheses:
[tex]\[ 3x + 6 = 2x + 2 \][/tex]
2. Subtract [tex]\(2x\)[/tex] from both sides to get the terms with [tex]\(x\)[/tex] on one side:
[tex]\[ 3x - 2x + 6 = 2 \][/tex]
[tex]\[ x + 6 = 2 \][/tex]
3. Subtract 6 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x + 6 - 6 = 2 - 6 \][/tex]
[tex]\[ x = -4 \][/tex]
So, the solution is [tex]\( x = -4 \)[/tex].
### Equation 3: [tex]\( \frac{2}{3}x = 4 \)[/tex]
1. To eliminate the fraction, multiply both sides by the reciprocal of [tex]\(\frac{2}{3}\)[/tex], which is [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[ \left(\frac{3}{2}\right) \left(\frac{2}{3}x\right) = 4 \left(\frac{3}{2}\right) \][/tex]
[tex]\[ x = 4 \times \frac{3}{2} \][/tex]
[tex]\[ x = \frac{12}{2} \][/tex]
[tex]\[ x = 6 \][/tex]
So, the solution is [tex]\( x = 6 \)[/tex].
### Equation 4: [tex]\( 1 - (x + 2) = 5 + x \)[/tex]
1. Distribute the negative sign inside the parentheses:
[tex]\[ 1 - x - 2 = 5 + x \][/tex]
[tex]\[ -x - 1 = 5 + x \][/tex]
2. Add [tex]\(x\)[/tex] to both sides to get all [tex]\(x\)[/tex] terms on one side:
[tex]\[ -x + x - 1 = 5 + x + x \][/tex]
[tex]\[ -1 = 5 + 2x \][/tex]
3. Subtract 5 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ -1 - 5 = 2x \][/tex]
[tex]\[ -6 = 2x \][/tex]
4. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{-6}{2} \][/tex]
[tex]\[ x = -3 \][/tex]
So, the solution is [tex]\( x = -3 \)[/tex].
In summary, the solutions to the equations are:
1. [tex]\( x = 1 \)[/tex]
2. [tex]\( x = -4 \)[/tex]
3. [tex]\( x = 6 \)[/tex]
4. [tex]\( x = -3 \)[/tex]
