Math worksheet for solving multi-step equations involving integers.

Here are the step-by-step solutions for each equation on the worksheet.

1) $-2(3n + 9) = 30$
* Step 1: Distribute the $-2$ into the parentheses.
$-6n - 18 = 30$
* Step 2: Add $18$ to both sides to isolate the term with $n$.
$-6n = 48$
* Step 3: Divide by $-6$.
$n = -8$

2) $9p + 9 - 5 = -12 + 6p$
* Step 1: Combine like terms on the left side ($9 - 5 = 4$).
$9p + 4 = -12 + 6p$
* Step 2: Subtract $6p$ from both sides to get variables on one side.
$3p + 4 = -12$
* Step 3: Subtract $4$ from both sides.
$3p = -16$
* Step 4: Divide by $3$.
$p = -\frac{16}{3}$ (or $-5 \frac{1}{3}$)

3) $18n - 6(-4n - 6) = 18 + 8n$
* Step 1: Distribute the $-6$ into the parentheses. Be careful with signs: $-6 \times -4n = +24n$ and $-6 \times -6 = +36$.
$18n + 24n + 36 = 18 + 8n$
* Step 2: Combine like terms on the left ($18n + 24n = 42n$).
$42n + 36 = 18 + 8n$
* Step 3: Subtract $8n$ from both sides.
$34n + 36 = 18$
* Step 4: Subtract $36$ from both sides.
$34n = -18$
* Step 5: Divide by $34$ and simplify the fraction.
$n = -\frac{18}{34} = -\frac{9}{17}$

4) $1 + 8p - 4p + 5 = 2(p - 3) - 2(p - 2)$
* Step 1: Simplify the left side ($8p - 4p = 4p$ and $1 + 5 = 6$).
$4p + 6 = 2(p - 3) - 2(p - 2)$
* Step 2: Distribute on the right side.
$4p + 6 = 2p - 6 - 2p + 4$
* Step 3: Combine like terms on the right side ($2p - 2p = 0$ and $-6 + 4 = -2$).
$4p + 6 = -2$
* Step 4: Subtract $6$ from both sides.
$4p = -8$
* Step 5: Divide by $4$.
$p = -2$

5) $-8x + 16 = -16x + 32$
* Step 1: Add $16x$ to both sides.
$8x + 16 = 32$
* Step 2: Subtract $16$ from both sides.
$8x = 16$
* Step 3: Divide by $8$.
$x = 2$

6) $16(5 - 4v) = -5 - v$
* Step 1: Distribute the $16$.
$80 - 64v = -5 - v$
* Step 2: Add $64v$ to both sides to make the variable positive.
$80 = -5 + 63v$
* Step 3: Add $5$ to both sides.
$85 = 63v$
* Step 4: Divide by $63$.
$v = \frac{85}{63}$ (This fraction cannot be simplified further)

7) $-12 = 2(2x + 4) - (7 - 3x)$
* Step 1: Distribute the $2$ and the negative sign (think of it as distributing $-1$).
$-12 = 4x + 8 - 7 + 3x$
* Step 2: Combine like terms on the right side ($4x + 3x = 7x$ and $8 - 7 = 1$).
$-12 = 7x + 1$
* Step 3: Subtract $1$ from both sides.
$-13 = 7x$
* Step 4: Divide by $7$.
$x = -\frac{13}{7}$ (or $-1 \frac{6}{7}$)

8) $-7r - 3r = -13 - 7 - r$
* Step 1: Combine like terms on both sides. Left: $-10r$. Right: $-20 - r$.
$-10r = -20 - r$
* Step 2: Add $r$ to both sides.
$-9r = -20$
* Step 3: Divide by $-9$. The negatives cancel out.
$r = \frac{20}{9}$ (or $2 \frac{2}{9}$)

9) $16 + 32q + 4 = -32q + 40$
* Step 1: Combine constants on the left ($16 + 4 = 20$).
$32q + 20 = -32q + 40$
* Step 2: Add $32q$ to both sides.
$64q + 20 = 40$
* Step 3: Subtract $20$ from both sides.
$64q = 20$
* Step 4: Divide by $64$ and simplify.
$q = \frac{20}{64} = \frac{5}{16}$

10) $-6x + 14 = 1 - 8x$
* Step 1: Add $8x$ to both sides.
$2x + 14 = 1$
* Step 2: Subtract $14$ from both sides.
$2x = -13$
* Step 3: Divide by $2$.
$x = -\frac{13}{2}$ (or $-6.5$)

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Final Answer:
1) $n = -8$
2) $p = -\frac{16}{3}$
3) $n = -\frac{9}{17}$
4) $p = -2$
5) $x = 2$
6) $v = \frac{85}{63}$
7) $x = -\frac{13}{7}$
8) $r = \frac{20}{9}$
9) $q = \frac{5}{16}$
10) $x = -\frac{13}{2}$