Answer :
Alright, let's go through these equations step-by-step to solve for [tex]\( x \)[/tex].
### 3.1.1 Solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ \frac{x}{8} + 9 = 2 \][/tex]
1. Subtract 9 from both sides to isolate the fraction:
[tex]\[ \frac{x}{8} = 2 - 9 \][/tex]
[tex]\[ \frac{x}{8} = -7 \][/tex]
2. Multiply both sides by 8 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = -7 \times 8 \][/tex]
[tex]\[ x = -56 \][/tex]
So, [tex]\( x \)[/tex] is [tex]\( -56 \)[/tex] in this equation.
### 3.1.25 Solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ x = 40 + 3x \][/tex]
1. Subtract [tex]\( 3x \)[/tex] from both sides to get the [tex]\( x \)[/tex] terms on one side:
[tex]\[ x - 3x = 40 \][/tex]
[tex]\[ -2x = 40 \][/tex]
2. Divide both sides by [tex]\(-2\)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{40}{-2} \][/tex]
[tex]\[ x = -20 \][/tex]
So, [tex]\( x \)[/tex] is [tex]\( -20 \)[/tex] in this equation.
### 3.1.38 Solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ 8^x = 32 \][/tex]
1. Rewrite both sides using a common base. Since [tex]\( 8 = 2^3 \)[/tex] and [tex]\( 32 = 2^5 \)[/tex], we can rewrite the equation as:
[tex]\[ (2^3)^x = 2^5 \][/tex]
[tex]\[ 2^{3x} = 2^5 \][/tex]
2. Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 3x = 5 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{5}{3} \][/tex]
So, [tex]\( x \)[/tex] is [tex]\( \frac{5}{3} \)[/tex] (or approximately [tex]\( 1.67 \)[/tex]) in this equation.
### 3.1.42 Solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ x - 3 = 14 \][/tex]
1. Add 3 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 14 + 3 \][/tex]
[tex]\[ x = 17 \][/tex]
So, [tex]\( x \)[/tex] is [tex]\( 17 \)[/tex] in this equation.
### 3.1.5 Solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ \frac{3x - 1}{2} = 4 \][/tex]
1. Multiply both sides by 2 to eliminate the denominator:
[tex]\[ 3x - 1 = 4 \times 2 \][/tex]
[tex]\[ 3x - 1 = 8 \][/tex]
2. Add 1 to both sides to isolate the [tex]\( 3x \)[/tex] term:
[tex]\[ 3x = 8 + 1 \][/tex]
[tex]\[ 3x = 9 \][/tex]
3. Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{9}{3} \][/tex]
[tex]\[ x = 3 \][/tex]
So, [tex]\( x \)[/tex] is [tex]\( 3 \)[/tex] in this equation.
In summary, the solutions for [tex]\( x \)[/tex] in the given equations are:
1. [tex]\( x = -56 \)[/tex]
2. [tex]\( x = -20 \)[/tex]
3. [tex]\( x = \frac{5}{3} \)[/tex] (or approximately [tex]\( 1.67 \)[/tex])
4. [tex]\( x = 17 \)[/tex]
5. [tex]\( x = 3 \)[/tex]
### 3.1.1 Solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ \frac{x}{8} + 9 = 2 \][/tex]
1. Subtract 9 from both sides to isolate the fraction:
[tex]\[ \frac{x}{8} = 2 - 9 \][/tex]
[tex]\[ \frac{x}{8} = -7 \][/tex]
2. Multiply both sides by 8 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = -7 \times 8 \][/tex]
[tex]\[ x = -56 \][/tex]
So, [tex]\( x \)[/tex] is [tex]\( -56 \)[/tex] in this equation.
### 3.1.25 Solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ x = 40 + 3x \][/tex]
1. Subtract [tex]\( 3x \)[/tex] from both sides to get the [tex]\( x \)[/tex] terms on one side:
[tex]\[ x - 3x = 40 \][/tex]
[tex]\[ -2x = 40 \][/tex]
2. Divide both sides by [tex]\(-2\)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{40}{-2} \][/tex]
[tex]\[ x = -20 \][/tex]
So, [tex]\( x \)[/tex] is [tex]\( -20 \)[/tex] in this equation.
### 3.1.38 Solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ 8^x = 32 \][/tex]
1. Rewrite both sides using a common base. Since [tex]\( 8 = 2^3 \)[/tex] and [tex]\( 32 = 2^5 \)[/tex], we can rewrite the equation as:
[tex]\[ (2^3)^x = 2^5 \][/tex]
[tex]\[ 2^{3x} = 2^5 \][/tex]
2. Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 3x = 5 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{5}{3} \][/tex]
So, [tex]\( x \)[/tex] is [tex]\( \frac{5}{3} \)[/tex] (or approximately [tex]\( 1.67 \)[/tex]) in this equation.
### 3.1.42 Solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ x - 3 = 14 \][/tex]
1. Add 3 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 14 + 3 \][/tex]
[tex]\[ x = 17 \][/tex]
So, [tex]\( x \)[/tex] is [tex]\( 17 \)[/tex] in this equation.
### 3.1.5 Solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ \frac{3x - 1}{2} = 4 \][/tex]
1. Multiply both sides by 2 to eliminate the denominator:
[tex]\[ 3x - 1 = 4 \times 2 \][/tex]
[tex]\[ 3x - 1 = 8 \][/tex]
2. Add 1 to both sides to isolate the [tex]\( 3x \)[/tex] term:
[tex]\[ 3x = 8 + 1 \][/tex]
[tex]\[ 3x = 9 \][/tex]
3. Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{9}{3} \][/tex]
[tex]\[ x = 3 \][/tex]
So, [tex]\( x \)[/tex] is [tex]\( 3 \)[/tex] in this equation.
In summary, the solutions for [tex]\( x \)[/tex] in the given equations are:
1. [tex]\( x = -56 \)[/tex]
2. [tex]\( x = -20 \)[/tex]
3. [tex]\( x = \frac{5}{3} \)[/tex] (or approximately [tex]\( 1.67 \)[/tex])
4. [tex]\( x = 17 \)[/tex]
5. [tex]\( x = 3 \)[/tex]
