Question

1. solve the equation $4x - 2(x - 2) = -9 + 5x - 8$.
2. use the distributive property to solve the equation $2(m + 2) = 22$. describe what it means to distribute the 2 to each term inside the parentheses.
3. what is peter’s number?

if you subtract 12 from my number and multiply the difference by -3, the result is -54.

1. higher order thinking use the distributive property to solve the equation $\frac{4x}{5} - x = \frac{x}{10} - \frac{9}{2}$.
2. how many solutions does the equation $-2(x + 4) = -2(x + 4) - 6$ have?
3. solve the equation $3(x + 4) = 2x + 4x - 6$ for $x$.

Explanation:

Problem 15: Solve \( 4x - 2(x - 2) = -9 + 5x - 8 \)

Step 1: Distribute the -2

Distribute \(-2\) into \( (x - 2) \):
\( 4x - 2x + 4 = -9 + 5x - 8 \)

Step 2: Simplify both sides

Simplify left side: \( 2x + 4 \)
Simplify right side: \( 5x - 17 \)
Equation becomes: \( 2x + 4 = 5x - 17 \)

Step 3: Subtract \( 2x \) from both sides

\( 4 = 3x - 17 \)

Step 4: Add 17 to both sides

\( 21 = 3x \)

Step 5: Divide by 3

\( x = 7 \)

Problem 16: Solve \( 2(m + 2) = 22 \) (and explain distribution)

Step 1: Distribute the 2

Distribute \( 2 \) into \( (m + 2) \):
\( 2m + 4 = 22 \) (Distributing means multiplying 2 by \( m \) and 2 by \( 2 \), so \( 2 \times m = 2m \) and \( 2 \times 2 = 4 \))

Step 2: Subtract 4 from both sides

\( 2m = 18 \)

Step 3: Divide by 2

\( m = 9 \)

Problem 17: Find Peter’s number

Let Peter’s number be \( n \). The problem states:
\( -3(n - 12) = -54 \) (Subtract 12 from \( n \), multiply by \(-3\), result is \(-54\))

Step 1: Divide both sides by -3

\( n - 12 = \frac{-54}{-3} = 18 \)

Step 2: Add 12 to both sides

\( n = 18 + 12 = 30 \)

Problem 18: Solve \( \frac{4x}{5} - x = \frac{x}{10} - \frac{9}{2} \)

Answer:

s:

1. \( \boldsymbol{x = 7} \)
2. \( \boldsymbol{m = 9} \) (Distribution: \( 2 \times m + 2 \times 2 \))
3. Peter’s number: \( \boldsymbol{30} \)
4. \( \boldsymbol{x = 15} \)
5. \( \boldsymbol{0} \) solutions
6. \( \boldsymbol{x = 6} \)