# Page 41

## 1. $$8-6x=13$$

<details> <summary> Example: </summary>

$$
5-7x=14
$$

Subtract 5 from both sides:

$$
-7x=14-5
$$

Simplify:

$$
-7x=9
$$

Divide both sides by -7:

$$
x=\frac{9}{-7}
$$

Simplify:

$$
x=-\frac{9}{7}
$$

</details>

## 2. $$9+3x=7-5x$$

<details> <summary> Example: </summary>

$$
7+2x=8-4x
$$

Add 4x to both sides:

$$
7+2x+4x=8
$$

Simplify:

$$
7+6x=8
$$

Subtract 7 from both sides:

$$
6x=8-7
$$

Simplify:

$$
6x=1
$$

Divide both sides by 6:

$$
x=\frac{1}{6}
$$

</details>

## 3. $$4+3(2x-1)=14$$

<details> <summary> Example: </summary>

$$
5+2(3x-1)=16
$$

First, distribute the 2 on the left side:

$$
5+6x-2=16
$$

Simplify:

$$
3+6x=16
$$

Subtract 3 from both sides:

$$
6x=16-3
$$

Simplify:

$$
6x=13
$$

Divide both sides by 6:

$$
x=\frac{13}{6}
$$

</details>

## 4. $$3+2(x-1)=4+3(x+1)-x$$

<details> <summary> Example: </summary>

To solve the equation:

$$2+3(x-4)=5+4(x+3)-x$$

First, distribute on both sides:

$$2+3x-12=5+4x+12-x$$

Simplify both sides:

$$3x-10=4x+17-x$$

Combine like terms:

$$3x-10=3x+17$$

Subtract $3x$ from both sides:

$$-10=17$$

Since this is a contradiction, the equation has no solution.

Thus, the solution is:

$$\text{No solution}$$

</details>

# Page 42

## 1. $$3(y-2)+7=6(2y-2)$$

<details> <summary> Example: </summary>

$$
4(y-3)+6=5(3y-2)
$$

First, distribute both sides:

$$
4y-12+6=15y-10
$$

Simplify:

$$
4y-6=15y-10
$$

Add 6 to both sides:

$$
4y=15y-4
$$

Subtract 15y from both sides:

$$
4y-15y=-4
$$

Simplify:

$$
-11y=-4
$$

Divide both sides by -11:

$$
y=\frac{-4}{-11}
$$

Simplify:

$$
y=\frac{4}{11}
$$

</details>

## 2. $$2(2t-3)+4t=8t-6$$

<details> <summary> Example: </summary>

$$
3(4t-5)+2t=14t-15
$$

First, distribute the 3 on the left side:

$$
12t-15+2t=14t-15
$$

Combine like terms on the left side:

$$
14t-15=14t-15
$$

Since both sides are identical, this equation is true for all values of \( t \). Thus, the solution is:

$$
t \in \mathbb{R}
$$

Or simply:

$$
\text{All real numbers.}
$$

</details>

## 3. $$\dfrac{1}{2}x+5=\dfrac{1}{3}x-\dfrac{7}{2}$$

<details> <summary> Example: </summary>

To solve the equation:

$$\dfrac{1}{3}x+6=\dfrac{1}{4}x-\dfrac{9}{2}$$

First, move all the $x$ terms to the left side and constants to the right side. Subtract $\dfrac{1}{4}x$ from both sides:

$$\dfrac{1}{3}x-\dfrac{1}{4}x+6=-\dfrac{9}{2}$$

Now, combine the $x$ terms. The common denominator for $\dfrac{1}{3}$ and $\dfrac{1}{4}$ is 12, so rewrite both fractions:

$$\dfrac{4}{12}x-\dfrac{3}{12}x+6=-\dfrac{9}{2}$$

Simplify:

$$\dfrac{1}{12}x+6=-\dfrac{9}{2}$$

Next, subtract 6 from both sides:

$$\dfrac{1}{12}x=-\dfrac{9}{2}-6$$

Rewrite 6 as $\dfrac{12}{2}$:

$$\dfrac{1}{12}x=-\dfrac{9}{2}-\dfrac{12}{2}$$

Simplify:

$$\dfrac{1}{12}x=-\dfrac{21}{2}$$

Now, divide both sides by $\dfrac{1}{12}$, which is the same as multiplying by 12 (keep, change, flip):

$$x=-\dfrac{21}{2} \times 12$$

Simplify:

$$x=-\dfrac{252}{2}$$

$$x=-126$$

Thus, the solution is:

$$x=-126$$

</details>

# Page 43

## 1. Find the zeroes of $$f(x)=\dfrac{5}{6}x+\dfrac{3}{4}$$

<details> <summary> Example: </summary>

To find the zeroes of the function:

$$
f(x)=\dfrac{4}{3}x+\dfrac{2}{5}
$$

Set $f(x) = 0$:

$$
0 = \dfrac{4}{3}x + \dfrac{2}{5}
$$

Subtract $\dfrac{2}{5}$ from both sides:

$$
-\dfrac{2}{5} = \dfrac{4}{3}x
$$

Multiply both sides by the reciprocal of $\dfrac{4}{3}$, which is $\dfrac{3}{4}$:

$$
x = -\dfrac{2}{5} \times \dfrac{3}{4}
$$

Multiply the fractions:

$$
x = -\dfrac{6}{20}
$$

Simplify:

$$
x = -\dfrac{3}{10}
$$

Thus, the zero of the function is:

$$
x = -\dfrac{3}{10}
$$

</details>

## 2. Find the zeroes of $$g(x)=-\dfrac{2}{5}x+\dfrac{5}{3}$$

<details> <summary> Example: </summary>

To find the zeroes of the function:

$$g(x)=-\dfrac{3}{4}x+\dfrac{7}{2}$$

Set $g(x)=0$:

$$0=-\dfrac{3}{4}x+\dfrac{7}{2}$$

Subtract $\dfrac{7}{2}$ from both sides:

$$-\dfrac{7}{2}=-\dfrac{3}{4}x$$

Multiply both sides by the reciprocal of $-\dfrac{3}{4}$, which is $-\dfrac{4}{3}$:

$$x=-\dfrac{7}{2}\times-\dfrac{4}{3}$$

Multiply the fractions:

$$x=\dfrac{28}{6}$$

Simplify:

$$x=\dfrac{14}{3}$$

Thus, the zero of the function is:

$$x=\dfrac{14}{3}$$

</details>

## 3. Find the zeroes of $$h(x)=-17$$

<details> <summary> Example: </summary>

To find the zeroes of the function:

$$
h(x) = 14
$$

Set $h(x) =$:

$$
0 = 14
$$

This is a contradiction, as 14 is never equal to 0. Therefore, the function $h(x) = 14$ has no zeroes.

Thus, the solution is:

$$
\text{No zeroes exist.}
$$

</details>

# Page 44

## 1. If $500 is invested in an account at an annual rate of 2% for 3 years, what is the final amount?

<details> <summary> Example: </summary>

To find the final amount using simple interest, we use the formula:

$$I=Prt$$

Where:
- $I$ is the interest.
- $P=300$ is the principal amount (initial investment).
- $r=0.06$ is the annual interest rate (6%).
- $t=8$ is the time in years.

Substitute the given values into the formula:

$$I=300\cdot0.06\cdot8$$

Simplify:

$$I=144$$

Now, to find the final amount $A$, add the interest to the principal:

$$A=P+I$$

$$A=300+144$$

Thus, the final amount is:

$$A=444$$

</details>

## 2. If $700 is invested in an account at an annual rate of 4% for 7 years, what is the final amount?

<details> <summary> Example: </summary>

To find the final amount using simple interest, we use the formula:

$$I=Prt$$

Where:
- $I$ is the interest.
- $P=800$ is the principal amount (initial investment).
- $r=0.05$ is the annual interest rate (5%).
- $t=6$ is the time in years.

Substitute the given values into the formula:

$$I=800\cdot0.05\cdot6$$

Simplify:

$$I=240$$

Now, to find the final amount $A$, add the interest to the principal:

$$A=P+I$$

$$A=800+240$$

Thus, the final amount is:

$$A=1040$$

</details>