Answer :
Alright, let's work through each part of the question in detail:
### 3.1 Solve for [tex]\(x\)[/tex]
#### 3.1.1 [tex]\(x(x-2)=8\)[/tex]
1. Expand the left side of the equation to get a quadratic equation:
[tex]\[ x^2 - 2x = 8 \][/tex]
2. Subtract 8 from both sides to set the equation to zero:
[tex]\[ x^2 - 2x - 8 = 0 \][/tex]
3. Factorize the quadratic equation:
[tex]\[ (x - 4)(x + 2) = 0 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
- [tex]\(x - 4 = 0\)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\(x = 4\)[/tex]
- [tex]\(x + 2 = 0\)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\(x = -2\)[/tex]
Therefore, the solutions are:
[tex]\[ x = 4 \text{ or } x = -2 \][/tex]
#### 3.1.2 [tex]\(3^{2x + 2} = \frac{1}{27}\)[/tex]
1. Recall that [tex]\(27 = 3^3\)[/tex], so [tex]\( \frac{1}{27} = 3^{-3} \)[/tex].
2. Set the exponents equal to each other:
[tex]\[ 2x + 2 = -3 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[ 2x + 2 = -3 \][/tex]
[tex]\[ 2x = -5 \][/tex]
[tex]\[ x = -\frac{5}{2} \][/tex]
However, the solution is:
[tex]\[ x = -2.5 \text{ (This also includes a complex component, } -2.5 + 2.85960086738013i\text{)} \][/tex]
### 3.2 Solve the following inequality and represent it graphically:
[tex]\[ -\frac{1}{3} < \frac{x-1}{6} < \frac{1}{18} \][/tex]
1. Split the compound inequality into two parts and solve each part separately.
#### Part 1:
[tex]\[ -\frac{1}{3} < \frac{x-1}{6} \][/tex]
2. Multiply both sides by 6 to clear the denominator:
[tex]\[ -2 < x - 1 \][/tex]
3. Add 1 to both sides:
[tex]\[ -1 < x \][/tex]
So, the first inequality is:
[tex]\[ x > -1 \][/tex]
#### Part 2:
[tex]\[ \frac{x-1}{6} < \frac{1}{18} \][/tex]
4. Multiply both sides by 18 to clear the denominator:
[tex]\[ 3(x - 1) < 1 \][/tex]
5. Divide both sides by 3:
[tex]\[ x - 1 < \frac{1}{3} \][/tex]
6. Add 1 to both sides:
[tex]\[ x < \frac{4}{3} \][/tex]
So, the second inequality is:
[tex]\[ x < \frac{4}{3} \][/tex]
Combining both parts, we get the solution for the compound inequality:
[tex]\[ -1 < x < \frac{4}{3} \][/tex]
The range of [tex]\(x\)[/tex] is:
[tex]\[ -1 < x < 1.33333333333333 \][/tex]
### 3.3 Determining the prices of yoghurts
#### 3.3.1 Assign variables:
Let:
- [tex]\(x\)[/tex] be the price of a fruit yoghurt.
- [tex]\(y\)[/tex] be the price of a plain yoghurt.
We are given two pieces of information:
1. A fruit yoghurt costs [tex]\(R 4\)[/tex] more than a plain yoghurt:
[tex]\[ x = y + 4 \][/tex]
2. Five fruit yoghurts and three plain yoghurts together cost [tex]\(R 84\)[/tex]:
[tex]\[ 5x + 3y = 84 \][/tex]
#### 3.3.2 Determine the individual prices of the yoghurts
1. We have the system of equations:
[tex]\[ x = y + 4 \][/tex]
[tex]\[ 5x + 3y = 84 \][/tex]
2. Substitute [tex]\(x = y + 4\)[/tex] into the second equation:
[tex]\[ 5(y + 4) + 3y = 84 \][/tex]
3. Expand and simplify:
[tex]\[ 5y + 20 + 3y = 84 \][/tex]
[tex]\[ 8y + 20 = 84 \][/tex]
[tex]\[ 8y = 64 \][/tex]
[tex]\[ y = 8 \][/tex]
4. Substitute [tex]\(y = 8\)[/tex] back into the first equation:
[tex]\[ x = y + 4 \][/tex]
[tex]\[ x = 8 + 4 \][/tex]
[tex]\[ x = 12 \][/tex]
Therefore, the individual prices are:
[tex]\[ \text{Price of fruit yoghurt } x = 12 \][/tex]
[tex]\[ \text{Price of plain yoghurt } y = 8 \][/tex]
### 3.1 Solve for [tex]\(x\)[/tex]
#### 3.1.1 [tex]\(x(x-2)=8\)[/tex]
1. Expand the left side of the equation to get a quadratic equation:
[tex]\[ x^2 - 2x = 8 \][/tex]
2. Subtract 8 from both sides to set the equation to zero:
[tex]\[ x^2 - 2x - 8 = 0 \][/tex]
3. Factorize the quadratic equation:
[tex]\[ (x - 4)(x + 2) = 0 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
- [tex]\(x - 4 = 0\)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\(x = 4\)[/tex]
- [tex]\(x + 2 = 0\)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\(x = -2\)[/tex]
Therefore, the solutions are:
[tex]\[ x = 4 \text{ or } x = -2 \][/tex]
#### 3.1.2 [tex]\(3^{2x + 2} = \frac{1}{27}\)[/tex]
1. Recall that [tex]\(27 = 3^3\)[/tex], so [tex]\( \frac{1}{27} = 3^{-3} \)[/tex].
2. Set the exponents equal to each other:
[tex]\[ 2x + 2 = -3 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[ 2x + 2 = -3 \][/tex]
[tex]\[ 2x = -5 \][/tex]
[tex]\[ x = -\frac{5}{2} \][/tex]
However, the solution is:
[tex]\[ x = -2.5 \text{ (This also includes a complex component, } -2.5 + 2.85960086738013i\text{)} \][/tex]
### 3.2 Solve the following inequality and represent it graphically:
[tex]\[ -\frac{1}{3} < \frac{x-1}{6} < \frac{1}{18} \][/tex]
1. Split the compound inequality into two parts and solve each part separately.
#### Part 1:
[tex]\[ -\frac{1}{3} < \frac{x-1}{6} \][/tex]
2. Multiply both sides by 6 to clear the denominator:
[tex]\[ -2 < x - 1 \][/tex]
3. Add 1 to both sides:
[tex]\[ -1 < x \][/tex]
So, the first inequality is:
[tex]\[ x > -1 \][/tex]
#### Part 2:
[tex]\[ \frac{x-1}{6} < \frac{1}{18} \][/tex]
4. Multiply both sides by 18 to clear the denominator:
[tex]\[ 3(x - 1) < 1 \][/tex]
5. Divide both sides by 3:
[tex]\[ x - 1 < \frac{1}{3} \][/tex]
6. Add 1 to both sides:
[tex]\[ x < \frac{4}{3} \][/tex]
So, the second inequality is:
[tex]\[ x < \frac{4}{3} \][/tex]
Combining both parts, we get the solution for the compound inequality:
[tex]\[ -1 < x < \frac{4}{3} \][/tex]
The range of [tex]\(x\)[/tex] is:
[tex]\[ -1 < x < 1.33333333333333 \][/tex]
### 3.3 Determining the prices of yoghurts
#### 3.3.1 Assign variables:
Let:
- [tex]\(x\)[/tex] be the price of a fruit yoghurt.
- [tex]\(y\)[/tex] be the price of a plain yoghurt.
We are given two pieces of information:
1. A fruit yoghurt costs [tex]\(R 4\)[/tex] more than a plain yoghurt:
[tex]\[ x = y + 4 \][/tex]
2. Five fruit yoghurts and three plain yoghurts together cost [tex]\(R 84\)[/tex]:
[tex]\[ 5x + 3y = 84 \][/tex]
#### 3.3.2 Determine the individual prices of the yoghurts
1. We have the system of equations:
[tex]\[ x = y + 4 \][/tex]
[tex]\[ 5x + 3y = 84 \][/tex]
2. Substitute [tex]\(x = y + 4\)[/tex] into the second equation:
[tex]\[ 5(y + 4) + 3y = 84 \][/tex]
3. Expand and simplify:
[tex]\[ 5y + 20 + 3y = 84 \][/tex]
[tex]\[ 8y + 20 = 84 \][/tex]
[tex]\[ 8y = 64 \][/tex]
[tex]\[ y = 8 \][/tex]
4. Substitute [tex]\(y = 8\)[/tex] back into the first equation:
[tex]\[ x = y + 4 \][/tex]
[tex]\[ x = 8 + 4 \][/tex]
[tex]\[ x = 12 \][/tex]
Therefore, the individual prices are:
[tex]\[ \text{Price of fruit yoghurt } x = 12 \][/tex]
[tex]\[ \text{Price of plain yoghurt } y = 8 \][/tex]
