Directions: Simplify, show all steps.

$7(2 x+1)-2(3 x+4)$
$2(x+3)+4(x-1)+10$

Applications:

The sum of an unknown number and 15 can be represented by the expression $n+15$. Give a binomial expression, in simplest form, that represents three times this quantity.
Eight times an unknown number which is then decreased by four is given by the expression $8 x-4$, where $x$ is the unknown number. Give an expression that represents one-fourth of this expression.

Problem 9: $7(2x+1)-2(3x+4)$
To simplify this expression, we apply the distributive property:
$7(2x) + 7(1) - 2(3x) - 2(4) = 14x + 7 - 6x - 8$.
Next, we combine the like terms:
$(14x - 6x) + (7 - 8) = 8x - 1$.

Problem 10: $2(x+3)+4(x-1)+10$
Again, apply the distributive property:
$2(x) + 2(3) + 4(x) - 4(1) + 10 = 2x + 6 + 4x - 4 + 10$.
Combining like terms gives:
$(2x + 4x) + (6 - 4 + 10) = 6x + 12$.

Problem 11: Represent three times the quantity $n+15$.
We represent this as:
$3(n+15) = 3(n) + 3(15) = 3n + 45$.

Problem 12: Represent one-fourth of the expression $8x-4$.
We write this as:
$\frac{1}{4}(8x-4) = \frac{1}{4}(8x) - \frac{1}{4}(4) = 2x - 1$.

Examples

Algebraic simplification is essential for modeling real-world scenarios. For example, if you are calculating the total cost of a shopping trip where you buy three shirts at $\$n+15$ each, the expression $3n+45$ quickly tells you your total cost. Similarly, if you have a budget of $8x-4$ dollars for four different departments, dividing that total into equal shares for each department, represented by $2x-1$, helps in efficient resource allocation. Mastering these steps ensures accuracy in daily financial planning and resource management.