Practice problems demonstrating the properties of equality in algebra.

Explanation:
Let’s go through each problem (6, 7, and 8) step by step, filling in the missing steps and justifying each one using properties of equality.

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Problem 6:
Equation:
1. $ x + (4x + 32) = 12 $ — Original equation
2. $ 5x + 32 = 12 $ — Associative property of addition (combine like terms: $x + 4x = 5x$)
3. $ 5x = -20 $ — Subtract 32 from both sides: $5x + 32 - 32 = 12 - 32$, so $5x = -20$. This uses the subtraction property of equality (if $a = b$, then $a - c = b - c$).
4. $ x = -4 $ — Divide both sides by 5: $ \frac{5x}{5} = \frac{-20}{5} $, so $x = -4$. This uses the division property of equality (if $a = b$ and $c \ne 0$, then $a/c = b/c$).

So the missing step is:
→ $ x = -4 $, justified by *Division property of equality*.

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Problem 7:
Equation:
1. $ 4(x - 6) = 40 $ — Original equation
2. $ x - 6 = 10 $ — Divide both sides by 4: $ \frac{4(x - 6)}{4} = \frac{40}{4} $. This uses the division property of equality.
3. $ x = 16 $ — Add 6 to both sides: $x - 6 + 6 = 10 + 6$, so $x = 16$. This uses the addition property of equality.

So the missing steps are:
- For row 2: *Division property of equality*
- For row 3: *Addition property of equality*

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Problem 8:
Equation:
1. $ 1.4 - 0.3x + 0.7x = 9.4 $ — Original equation
2. $ 1.4 + 0.4x = 9.4 $ — Combine like terms: $-0.3x + 0.7x = 0.4x$. This uses the commutative and associative properties of addition, or simply combining like terms (which relies on those properties). But since the instruction is about *properties of equality*, this step is algebraic simplification, not a property of equality yet. So the justification here is usually just “combine like terms” — but if we must name a property, it's the associative property of addition (re-grouping) and commutative property of addition (reordering), though many textbooks accept “combine like terms” as sufficient. However, the next steps involve equality properties.

3. $ 0.4x = 8 $ — Subtract 1.4 from both sides: $1.4 + 0.4x - 1.4 = 9.4 - 1.4$, so $0.4x = 8$. This is the subtraction property of equality.

4. $ x = 20 $ — Divide both sides by 0.4: $ \frac{0.4x}{0.4} = \frac{8}{0.4} $, so $x = 20$. This is the division property of equality.

So the missing justifications:
- Row 2: *Combine like terms* (or more precisely, associative + commutative properties of addition — but for this level, “combine like terms” is acceptable; however, since the table expects *properties of equality*, and this step doesn’t change equality (just rewrites same side), it’s not strictly a property of equality — but the problem likely expects “combine like terms” or leaves blank. Since other rows use equality properties, let’s assume they want:
- Row 2: *Simplify (combine like terms)* — but if only equality properties allowed, maybe it's omitted. However, looking at problem 6, they labeled “Associative property of addition” for combining $x + 4x$, which is actually *not* associative — it’s combining like terms via distributive/identity. Hmm — but we follow the pattern used in #6.

In #6, they said $x + (4x + 32) = 12$ → $5x + 32 = 12$ is “Associative property of addition”. That’s technically inaccurate (associative is about grouping, not combining coefficients), but since the worksheet uses that label, we’ll match their style.

So for #8, step 2: $-0.3x + 0.7x = 0.4x$ — they may expect “Associative property of addition” again, or “Combining like terms**”. But to stay consistent with #6, where they used “Associative property of addition” for regrouping and combining, we’ll use:
→ *Associative property of addition* (even though slightly misapplied — but matching worksheet style)

Then:
- Row 3: *Subtraction property of equality*
- Row 4: *Division property of equality*

Now, let’s write all missing entries clearly:

Problem 6, last row:
Equation: $ x = -4 $
Steps: Division property of equality

Problem 7, second row:
Equation: $ x - 6 = 10 $
Steps: Division property of equality

Problem 7, third row:
Equation: $ x = 16 $
Steps: Addition property of equality

Problem 8, second row:
Equation: $ 1.4 + 0.4x = 9.4 $
Steps: Associative property of addition *(matching #6’s style)*

Problem 8, third row:
Equation: $ 0.4x = 8 $
Steps: Subtraction property of equality

Problem 8, fourth row:
Equation: $ x = 20 $
Steps: Division property of equality

Let me double-check calculations:

- #6: $x + 4x + 32 = 12$ → $5x = -20$ → $x = -4$ ✔
- #7: $4(x - 6) = 40$ → divide by 4: $x - 6 = 10$ → add 6: $x = 16$ ✔
- #8: $1.4 - 0.3x + 0.7x = 9.4$ → $1.4 + 0.4x = 9.4$ → subtract 1.4: $0.4x = 8$ → divide by 0.4: $x = 20$ ✔
(since $8 ÷ 0.4 = 80 ÷ 4 = 20$) ✔

All correct.

Final Answer:
6. $x = -4$, Division property of equality
7. $x - 6 = 10$, Division property of equality; $x = 16$, Addition property of equality
8. $1.4 + 0.4x = 9.4$, Associative property of addition; $0.4x = 8$, Subtraction property of equality; $x = 20$, Division property of equality