I'll walk you through each part of this worksheet. Let's solve each section one by one.

Problem 4-9

Solve for $x$.

a.

$-2(4 - 3x) - 6x = 10$

1. Distribute $-2$: $-8 + 6x - 6x = 10$
2. Simplify terms: $-8 = 10$ Since this statement is false, there is no solution for part (a).

b.

$\frac{x - 5}{-2} = \frac{x - 1}{-3}$

1. Cross-multiply: $-3(x - 5) = -2(x - 1)$
2. Distribute on both sides: $-3x + 15 = -2x + 2$
3. Rearrange terms to isolate $x$: $-x = -13$ $x = 13$ So, the solution for part (b) is $x = 13$.

Problem 4-12

Find the dimensions of the generic rectangles and write the area as a product and a sum.

1. First Rectangle:

   • The area given is $3y^2 - 12y$.
   • Factor out $3y$: $3y(y - 4)$
   • Dimensions: $3y$ and $y - 4$
   • Area as a product: $3y(y - 4)$
   • Area as a sum: $3y^2 - 12y$

2. Second Rectangle:

   • The area given is $3y^2 - 20$.
   • Factor out if possible (but no common factors are visible here).
   • Dimensions: Left as is since it doesn't factor further with whole numbers.

Problem 4-14

Write an equation to solve for the dimensions of a rectangle with a perimeter of 30 inches.

1. Let the width be $w$.
2. Then, the length is $3w - 1$.
3. The perimeter $P$ is given by: $2w + 2(3w - 1) = 30$
4. Expand and solve for $w$: $2w + 6w - 2 = 30$ $8w = 32$ $w = 4$
5. Substitute $w = 4$ to find the length: $\text{Length} = 3(4) - 1 = 11$ So, the dimensions are 4 inches (width) and 11 inches (length).

Problem 4-15

This question is about setting up a new roster. It asks to substitute Barbara for Allison and Lakeisha for Melinda.

Starting Roster for the game:

• Samantha
• Carly
• Barbara (substitute for Allison)
• Kendra
• Lakeisha (substitute for Melinda)

Would you like further details on any specific part of these solutions, or do you have any additional questions?

Further Practice Questions

1. Solve for $x$ in equations involving distribution, like $-3(2x + 4) + 5x = 12$.
2. Factor and find the dimensions for a generic rectangle with area $4x^2 - 16x$.
3. Calculate the perimeter of a rectangle given different length-width relationships, such as length being twice the width.
4. Rewrite a sum of terms into a factored form, like $x^2 - 9x + 14$.
5. Create a roster with substitutions based on different constraints.

Tip

When solving equations, always check if you can simplify or factor out terms before expanding. This often makes calculations easier and reduces errors.