Here are the step-by-step solutions for each problem on the worksheet.
1. $14.13 = x - 4.25$
To get $x$ by itself, add $4.25$ to both sides.
$$14.13 + 4.25 = 18.38$$
So, $x = 18.38$.
2. $14.1 + x = 43.9$
Subtract $14.1$ from both sides to isolate $x$.
$$43.9 - 14.1 = 29.8$$
So, $x = 29.8$.
3. $4x - 10 = x - 7$
First, subtract $x$ from both sides to group the $x$'s:
$$3x - 10 = -7$$
Next, add $10$ to both sides:
$$3x = 3$$
Divide by $3$:
$$x = 1$$
4. $2 - x = 5x - 10$
Add $x$ to both sides:
$$2 = 6x - 10$$
Add $10$ to both sides:
$$12 = 6x$$
Divide by $6$:
$$x = 2$$
5. $2 + x = -5x - 22$
Add $5x$ to both sides:
$$2 + 6x = -22$$
Subtract $2$ from both sides:
$$6x = -24$$
Divide by $6$:
$$x = -4$$
6. $2 + 4x = 2x + 20$
Subtract $2x$ from both sides:
$$2 + 2x = 20$$
Subtract $2$ from both sides:
$$2x = 18$$
Divide by $2$:
$$x = 9$$
7. $-1 + 4x = 6x - 7$
Subtract $4x$ from both sides:
$$-1 = 2x - 7$$
Add $7$ to both sides:
$$6 = 2x$$
Divide by $2$:
$$x = 3$$
8. $\frac{42x^3}{2x}$
Divide the numbers: $42 \div 2 = 21$.
Divide the variables: $x^3 \div x = x^2$.
Result: $21x^2$.
9. $\frac{52x^2}{2x}$
Divide the numbers: $52 \div 2 = 26$.
Divide the variables: $x^2 \div x = x$.
Result: $26x$.
10. $\frac{x^2 + 7x + 6}{x + 1}$
Factor the top part (numerator). We need two numbers that multiply to $6$ and add to $7$. Those numbers are $6$ and $1$.
So, $x^2 + 7x + 6 = (x + 6)(x + 1)$.
Now cancel out $(x + 1)$ from the top and bottom.
Result: $x + 6$.
11. $\frac{x + 2}{x^2 + 9x + 18}$
Factor the bottom part (denominator). We need two numbers that multiply to $18$ and add to $9$. Those numbers are $6$ and $3$.
So, $x^2 + 9x + 18 = (x + 6)(x + 3)$.
The expression becomes $\frac{x + 2}{(x + 6)(x + 3)}$. Nothing cancels out.
Result: $\frac{x + 2}{(x + 6)(x + 3)}$ or $\frac{x + 2}{x^2 + 9x + 18}$.
12. $\frac{x^2 + 8x + 15}{x^2 + 6x + 8}$
Factor the top: Numbers that multiply to $15$ and add to $8$ are $5$ and $3$. So, $(x + 5)(x + 3)$.
Factor the bottom: Numbers that multiply to $8$ and add to $6$ are $4$ and $2$. So, $(x + 4)(x + 2)$.
Nothing cancels out between the top and bottom.
Result: $\frac{(x + 5)(x + 3)}{(x + 4)(x + 2)}$ or $\frac{x^2 + 8x + 15}{x^2 + 6x + 8}$.
Final Answer:
1. $x = 18.38$
2. $x = 29.8$
3. $x = 1$
4. $x = 2$
5. $x = -4$
6. $x = 9$
7. $x = 3$
8. $21x^2$
9. $26x$
10. $x + 6$
11. $\frac{x + 2}{(x + 6)(x + 3)}$
12. $\frac{(x + 5)(x + 3)}{(x + 4)(x + 2)}$
Parent Tip: Review the logic above to help your child master the concept of basic algebra practice worksheet.
