Algebra worksheet for solving equations and simplifying expressions.

Let’s solve each problem one by one. I’ll show my work carefully so you can follow along.

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1. 14.13 = x - 4.25

Add 4.25 to both sides:

x = 14.13 + 4.25
x = 18.38

✔ Check: 18.38 - 4.25 = 14.13 → Correct.

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2. 14.1 + x = 43.9

Subtract 14.1 from both sides:

x = 43.9 - 14.1
x = 29.8

✔ Check: 14.1 + 29.8 = 43.9 → Correct.

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3. 4x - 10 = x - 7

Subtract x from both sides:

3x - 10 = -7

Add 10 to both sides:

3x = 3

Divide by 3:

x = 1

✔ Check: Left side: 4(1) - 10 = -6; Right side: 1 - 7 = -6 → Correct.

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4. 2 - x = 5x - 10

Add x to both sides:

2 = 6x - 10

Add 10 to both sides:

12 = 6x

Divide by 6:

x = 2

✔ Check: Left: 2 - 2 = 0; Right: 5(2) - 10 = 0 → Correct.

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5. 2 + x = -5x - 22

Add 5x to both sides:

2 + 6x = -22

Subtract 2:

6x = -24

Divide by 6:

x = -4

✔ Check: Left: 2 + (-4) = -2; Right: -5(-4) - 22 = 20 - 22 = -2 → Correct.

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6. 2 + 4x = 2x + 20

Subtract 2x from both sides:

2 + 2x = 20

Subtract 2:

2x = 18

Divide by 2:

x = 9

✔ Check: Left: 2 + 4(9) = 38; Right: 2(9) + 20 = 38 → Correct.

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7. -1 + 4x = 6x - 7

Subtract 4x from both sides:

-1 = 2x - 7

Add 7:

6 = 2x

Divide by 2:

x = 3

✔ Check: Left: -1 + 4(3) = 11; Right: 6(3) - 7 = 11 → Correct.

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Now for the algebraic simplifications (problems 8–12). These are not equations to solve — they’re expressions to simplify.

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8. (42x³)/(2x)

Divide coefficients: 42 ÷ 2 = 21

Subtract exponents on x: x³ ÷ x¹ = x²

So: 21x²

✔ Check: 2x × 21x² = 42x³ → Correct.

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9. (52x²)/(2x)

52 ÷ 2 = 26

x² ÷ x = x

So: 26x

✔ Check: 2x × 26x = 52x² → Correct.

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10. (x² + 7x + 6)/(x + 1)

Factor numerator: Look for two numbers that multiply to 6 and add to 7 → 6 and 1.

So: x² + 7x + 6 = (x + 6)(x + 1)

Now: [(x + 6)(x + 1)] / (x + 1) = x + 6 (as long as x ≠ -1)

✔ Check: Multiply (x + 6)(x + 1) = x² + 7x + 6 → Correct.

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11. (x + 2)/(x² + 9x + 18)

Factor denominator: Find two numbers that multiply to 18 and add to 9 → 6 and 3.

So: x² + 9x + 18 = (x + 6)(x + 3)

Numerator is (x + 2), which doesn’t factor with denominator.

No common factors → expression is already simplified.

Answer: (x + 2)/[(x + 6)(x + 3)]

But let’s double-check if we made a mistake — maybe it was meant to be factored differently? No, x+2 doesn’t match any factor in denominator. So yes, this is simplest form.

Wait — perhaps the problem expects us to leave it as is? Or did I misread?

Actually, looking again — no cancellation possible. So final answer is just the fraction as given, but factored denominator.

But since the instruction says “solve the equations” — wait, these aren’t equations! They’re expressions. Maybe the worksheet meant “simplify”.

In any case, for #11, no simplification beyond factoring denominator.

But let me check if numerator matches any factor — x+2 vs x+6 or x+3 → no.

So: (x + 2)/[(x + 6)(x + 3)]

Alternatively, leave as (x+2)/(x²+9x+18) — but factored is better.

I think the expected answer is to factor and see if anything cancels — nothing does.

So we write: (x + 2)/[(x + 6)(x + 3)]

But actually, in many textbooks, they’d just say “cannot be simplified further”, but since it’s listed under “solve”, maybe it’s a trick? No — probably just simplify.

Wait — let me recheck problem 11: (x+2)/(x²+9x+18)

Yes, denominator factors to (x+3)(x+6), numerator is prime relative to that.

So answer is: (x + 2)/[(x + 3)(x + 6)]

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12. (x² + 8x + 15)/(x² + 6x + 8)

Factor numerator: x² + 8x + 15 → factors of 15 that add to 8: 5 and 3 → (x+5)(x+3)

Factor denominator: x² + 6x + 8 → factors of 8 that add to 6: 4 and 2 → (x+4)(x+2)

So: [(x+5)(x+3)] / [(x+4)(x+2)]

No common factors → cannot simplify.

Wait — is that right? Let me check:

Numerator: (x+5)(x+3) = x² + 8x + 15 ✔️

Denominator: (x+4)(x+2) = x² + 6x + 8 ✔️

No overlapping factors → so expression is already simplified.

But wait — maybe I made a mistake? Let me see if there's another way.

No — correct.

So answer: (x+5)(x+3)/[(x+4)(x+2)]

Or leave as is — but factored form is preferred.

Actually, in some contexts, they might expect you to write it as is, but since others simplified, perhaps this one doesn't reduce.

Wait — let me double-check problem 12: (x² + 8x + 15)/(x² + 6x + 8)

Yes, no common binomial factors.

So final simplified form is: (x+3)(x+5)/[(x+2)(x+4)]

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Now, compiling all answers:

Final Answer:
1. x = 18.38
2. x = 29.8
3. x = 1
4. x = 2
5. x = -4
6. x = 9
7. x = 3
8. 21x²
9. 26x
10. x + 6
11. (x + 2)/[(x + 3)(x + 6)]
12. (x + 3)(x + 5)/[(x + 2)(x + 4)]