Here are the solutions for each section of the worksheet:

Warm-Up:

Select all the expressions represented by the tape diagram.

The tape diagram shows three equal parts labeled $x+5$, which suggests multiplication:

• This represents $3(x+5)$.

Correct selections: ✅ $3(x+5)$
✅ $(x+5) \cdot 3$
❌ $3(x+15)$ (incorrect because it adds 15 instead of 5 three times)
❌ $15 + 3x$ (incorrect because it does not follow the multiplication structure)

Practice Section

Solving Equations:

1. 15x - 10 = 65

   • $15x = 65 + 10$
   • $15x = 75$
   • $x = \frac{75}{15} = 5$

2. 3(x + 7) = -12

   • Divide by 3: $x + 7 = \frac{-12}{3}$
   • $x + 7 = -4$
   • $x = -4 - 7 = -11$

3. -100x - 100 = 0

   • Add 100: $-100x = 100$
   • Divide by -100: $x = \frac{100}{-100} = -1$

Solve Each Equation:

1. -4x = -28

   • Divide by -4: $x = \frac{-28}{-4} = 7$

2. -4(x + 1) = -28

   • Divide by -4: $x + 1 = \frac{-28}{-4} = 7$
   • $x = 7 - 1 = 6$

3. x + 7 = -1

   • Subtract 7: $x = -1 - 7 = -8$

4. -3x + 7 = -1

   • Subtract 7: $-3x = -1 - 7 = -8$
   • Divide by -3: $x = \frac{-8}{-3} = \frac{8}{3}$

Match Each Story to an Equation:

1. A stack of nested paper cups is 8 inches tall. The first cup is 4 inches tall, and each additional cup adds $\frac{1}{4}$ inch.

   • Correct equation: $\frac{1}{4} + 4x = 8$ (A)

2. A baker uses 4 cups of flour. She uses $\frac{1}{4}$ cup to flour the counters and the rest to make 8 muffins.

   • Correct equation: $4 + \frac{1}{4} x = 8$ (B)

3. Mariana has an 8-foot ribbon. She cuts off $\frac{1}{4}$ foot and divides the remainder into 4 equal pieces.

   • Correct equation: $8x + \frac{1}{4} = 4$ (C)

Would you like further explanation on any part?

Here are 5 related questions for practice:

1. Solve $2(x - 3) = 14$.
2. If $5x + 2 = 3x + 10$, what is $x$?
3. Translate the phrase "three times a number decreased by 7 equals 11" into an equation.
4. Solve for $x$: $\frac{x}{4} + 2 = 6$.
5. A number $x$ is doubled and then reduced by 5 to give 13. Find $x$.

Tip: Always isolate the variable step by step using inverse operations.