# 2.6: Exercise 2

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State the property or indicate that no property is applicable.

1. $$5+x=x+5$$
2. $$7(a-2)=7a-7(2)$$
3. $$5(x\cdot 2)=(5x)(2)$$
4. $$(y+2)+4=4+(y+2)$$
5. $$6\cdot (t\cdot 2)=(6\cdot t)\cdot (6\cdot 2)$$
6. $$w-5=5-w$$
7. $$xy=yx$$
8. $$a\div \pi=\pi \div a$$
9. $$(-4.7)+[(6.5)+(\sqrt{2})]=[(-4.7)+(6.5)]+(\sqrt{2})$$
10. $$x-4=4-x$$
11. Factor $$12x+18$$
12. Factor $$24x^2+12x$$

The solutions follow:

Keep them covered up till you have worked out each of the problems above.

1. $$5+x=x+5$$
Solution:
$$5+x=x+5$$ illustrates the commutative property of addition. The terms commuted to each other’s position.
2. $$7(a-2)=7a-7(2)$$
Solution:
$$7(a-2)=7a-7(2)$$ illustrates the distributive property of multiplication over addition.
3. $$5(x\cdot 2)=(5x)(2)$$
Solution:
$$5(x\cdot 2)=(5x)(2)$$ illustrates the associative property of multiplication.
4. $$(y+2)+4=4+(y+2)$$
Solution:
$$(y+2)+4=4+(y+2)$$ illustrates the commutative property of addition. $$(y+2)$$ and 4 have commuted to each other’s position. $$y$$ remains grouped with 2.

N regrouping took place.

5. $$6\cdot (t\cdot 2)=(6\cdot t)\cdot (6\cdot 2)$$
Solution:
$$6\cdot (t\cdot 2)=(6t)\cdot (6\cdot 2)$$ does not illustrate any of the mentioned properties. You cannot distribute multiplication over multiplication.
6. $$w-5=5-w$$
Solution:
$$w-5=5-w$$ does not illustrate any of the mentioned properties. Subtraction is not commutative.
We shall learn later that $$w-5=w+(-5)=(-5)+w$$ because addition is commutative.
7. $$xy=yx$$
Solution:
$$xy=yx$$ illustrates the commutative property of multiplication. The terms commuted to each other’s position.
8. $$a\div \pi=\pi \div a$$
Solution:
$$a\div \pi=\pi \div a$$ does not illustrate any of the mentioned properties. Division is not commutative.
9. $$(-4.7)+[(6.5)+(\sqrt{2})]=[(-4.7)+(6.5)]+(\sqrt{2})$$
Solution:
$$(-4.7)+[(6.5)+(\sqrt{2})]=[(-4.7)+(6.5)]+(\sqrt{2})$$ illustrates the associative property of addition. $$6.5$$ is associated (grouped) with $$\sqrt{2}$$ on the left and $$-4.7$$ on the right.
10. $$x-4=4-x$$
Solution:
$$x-4=4-x$$ does not illustrate any of the mentioned properties. Subtraction is not commutative.
11. Factor $$12x+18$$
Solution:
$$\begin{array}{rcl lll} 12x+18&=&(3)(4)x+(3)(6)\\ &=&3(4x+6)\\ \end{array}$$

A quick check: $$3(4x+6)=3(4x)+3(6)=12x+18$$Gthe result grees wiht the iginal statement.

12. Factor $$24x^2+12x$$
Solution:
$$\begin{array}{rcl lll} 24x^2+12x&=&2(12)x\ x+1(12)x\\ &=&12x(2x+1)\\ \end{array}$$
Note the $$1$$ besides the $$12$$.

A quick check: $$12x(2x+1)=12x(2x)+12x(1)=24x^2+12x$$

2.6: Exercise 2 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.