Worksheet for solving multi-step equations involving integers.

Let’s solve each equation step by step. I’ll go one at a time, showing all the work clearly so you can follow along easily.

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Problem 1: -2(3n + 9) = 30

Step 1: Distribute the -2
→ -2 × 3n = -6n
→ -2 × 9 = -18
So: -6n - 18 = 30

Step 2: Add 18 to both sides to isolate the variable term
→ -6n = 30 + 18
→ -6n = 48

Step 3: Divide both sides by -6
→ n = 48 ÷ (-6)
→ n = -8

✔ Check: Plug back in: -2(3×(-8) + 9) = -2(-24 + 9) = -2(-15) = 30 → Correct!

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Problem 2: 9p + 9 - 5 = -12 + 6p

Step 1: Simplify left side: 9 - 5 = 4
→ 9p + 4 = -12 + 6p

Step 2: Subtract 6p from both sides
→ 3p + 4 = -12

Step 3: Subtract 4 from both sides
→ 3p = -16

Step 4: Divide by 3
→ p = -16/3

Wait — let me double-check that. The problem says “Integers” in the title, but maybe it’s okay if answer is fraction? Let’s verify:

Left: 9*(-16/3) + 4 = -48 + 4 = -44
Right: -12 + 6*(-16/3) = -12 - 32 = -44 → Correct.

But since the worksheet says “Integers”, perhaps I made a mistake? Let me re-read:

Original: 9p + 9 - 5 = -12 + 6p → yes, simplifies correctly. Maybe integer refers to coefficients, not necessarily solution? We’ll keep it as is.

Actually, wait — maybe I misread the problem? Let me check again.

No, it’s correct. So p = -16/3 is right, even if not integer. But let’s hold on — maybe there’s an error in my reading?

Looking back at image: Problem 2 is “9p + 9 - 5 = -12 + 6p” — yes.

Hmm. Perhaps the worksheet allows non-integers? Or maybe I should write as mixed number? No, improper fraction is fine.

But let’s continue and see others — maybe this one is intentional.

Actually, let me recalculate:

9p + 4 = -12 + 6p
Subtract 6p: 3p + 4 = -12
Subtract 4: 3p = -16
Divide: p = -16/3 → Yes.

Okay, moving on.

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Problem 3: 18n - 6(-4n - 6) = 18 + 8n

Step 1: Distribute the -6 across (-4n - 6)
→ -6 × -4n = +24n
→ -6 × -6 = +36
So: 18n + 24n + 36 = 18 + 8n

Step 2: Combine like terms on left
→ 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
→ n = -18/34 → simplify by dividing numerator and denominator by 2
→ n = -9/17

Check: Left: 18*(-9/17) -6*(-4*(-9/17)-6)
First, inside: -4*(-9/17)=36/17; minus 6 = 36/17 - 102/17 = -66/17
Then -6 * (-66/17) = 396/17
18*(-9/17) = -162/17
Total left: -162/17 + 396/17 = 234/17

Right: 18 + 8*(-9/17) = 18 - 72/17 = 306/17 - 72/17 = 234/17 → Correct.

Again, fractional. Okay.

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Problem 4: 1 + 8p - 4p + 5 = 2(p - 3) - 2(p - 2)

Step 1: Simplify left side: combine constants and p terms
1 + 5 = 6; 8p - 4p = 4p → 4p + 6

Step 2: Right side: distribute both 2s
2(p - 3) = 2p - 6
-2(p - 2) = -2p + 4
Combine: 2p - 6 - 2p + 4 = (2p - 2p) + (-6 + 4) = 0p - 2 = -2

So now: 4p + 6 = -2

Step 3: Subtract 6 from both sides
→ 4p = -8

Step 4: Divide by 4
→ p = -2

Check: Left: 1 + 8*(-2) -4*(-2) +5 = 1 -16 +8 +5 = (1+8+5) -16 = 14 -16 = -2
Right: 2(-2-3) -2(-2-2) = 2(-5) -2(-4) = -10 +8 = -2 → Correct.

Good, integer answer.

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Problem 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

Check: Left: -8*2 +16 = -16+16=0
Right: -16*2 +32 = -32+32=0 → Correct.

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Problem 6: 16(5 - 4v) = -5 - v

Step 1: Distribute 16
→ 80 - 64v = -5 - v

Step 2: Add 64v to both sides
→ 80 = -5 + 63v

Step 3: Add 5 to both sides
→ 85 = 63v

Step 4: Divide by 63
→ v = 85/63

Can we simplify? 85 and 63 share no common factors (85=5×17, 63=7×9), so leave as is.

Check: Left: 16(5 - 4*(85/63)) = 16(5 - 340/63) = 16(315/63 - 340/63) = 16(-25/63) = -400/63
Right: -5 - 85/63 = -315/63 - 85/63 = -400/63 → Correct.

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Problem 7: -12 = 2(2x + 4) - (7 - 3x)

Step 1: Distribute
2(2x + 4) = 4x + 8
-(7 - 3x) = -7 + 3x

So right side: 4x + 8 -7 + 3x = 7x + 1

Equation: -12 = 7x + 1

Step 2: Subtract 1 from both sides
→ -13 = 7x

Step 3: Divide by 7
→ x = -13/7

Check: Right: 2(2*(-13/7)+4) - (7 - 3*(-13/7))
First part: 2*(-26/7 + 28/7) = 2*(2/7) = 4/7
Second part: -(7 + 39/7) = -(49/7 + 39/7) = -88/7
Total: 4/7 - 88/7 = -84/7 = -12 → Correct.

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Problem 8: -7r - 3r = -13 - 7 - r

Step 1: Combine like terms on left: -7r -3r = -10r
Right: -13 -7 = -20, so -20 - r

Equation: -10r = -20 - r

Step 2: Add r to both sides
→ -9r = -20

Step 3: Divide by -9
→ r = 20/9

Check: Left: -7*(20/9) -3*(20/9) = -140/9 -60/9 = -200/9
Right: -13 -7 -20/9 = -20 -20/9 = -180/9 -20/9 = -200/9 → Correct.

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Problem 9: 16 + 32q + 4 = -32q + 40

Step 1: Combine constants on left: 16+4=20
→ 20 + 32q = -32q + 40

Step 2: Add 32q to both sides
→ 20 + 64q = 40

Step 3: Subtract 20 from both sides
→ 64q = 20

Step 4: Divide by 64
→ q = 20/64 = 5/16 (simplified)

Check: Left: 16 + 32*(5/16) +4 = 16 + 10 +4 = 30
Right: -32*(5/16) +40 = -10 +40 = 30 → Correct.

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Problem 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 = -13/2

Check: Left: -6*(-13/2) +14 = 78/2 +14 = 39 +14 = 53
Right: 1 -8*(-13/2) = 1 + 52 = 53 → Correct.

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Now, compiling all answers:

1. n = -8
2. p = -16/3
3. n = -9/17
4. p = -2
5. x = 2
6. v = 85/63
7. x = -13/7
8. r = 20/9
9. q = 5/16
10. x = -13/2

Even though the worksheet says “Integers”, some solutions are fractions. That’s mathematically correct based on the equations given.

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}