Alright, let's tackle each part of the question one by one and identify the properties that justify each of them.
### i) -x + 9 = 9 + x
This equation demonstrates the Commutative Property of Addition. According to this property, the order in which we add numbers does not change the sum. For example, [tex]\(a + b = b + a\)[/tex]. Here, [tex]\(-x + 9 = 9 + x\)[/tex].
### ii) 2(x + 3) = 2x + 6
This equation uses the Distributive Property. The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. This is shown as [tex]\( a(b + c) = ab + ac \)[/tex]. In this case, [tex]\(2(x + 3) = 2x + 6\)[/tex].
### iii) [tex]\(xy(1) = 5y\)[/tex]
This expression demonstrates the Multiplicative Identity Property. The multiplicative identity property states that any number multiplied by 1 remains unchanged, thus [tex]\( a \times 1 = a \)[/tex]. Here, it is showing that 1 multiplied by any expression (in this case, [tex]\(xy\)[/tex]) remains [tex]\(xy\)[/tex], implying the property of multiplicative identity. [tex]\( (xy)1 = xy \)[/tex], but since it equals [tex]\(5y\)[/tex], it suggests [tex]\(xy = 5y \)[/tex].
### iv) If [tex]\(8 + 2 < 14 < 20\)[/tex], then [tex]\(8 + 2 < 20\)[/tex]
This statement utilizes the Transitive Property of Inequality. According to this property, if [tex]\(a < b\)[/tex] and [tex]\(b < c\)[/tex], then [tex]\(a < c\)[/tex]. Here, [tex]\(8 + 2 < 14\)[/tex] and [tex]\(14 < 20\)[/tex], so it implies [tex]\(8 + 2 < 20\)[/tex].
### v) If [tex]\((m - n) < (p + q)\)[/tex] and [tex]\(r > 0\)[/tex], then [tex]\((m - n)r < (p + q)r\)[/tex]
This statement shows the Multiplication Property of Inequality. This property states that if you multiply both sides of an inequality by a positive number, the inequality remains the same. Since [tex]\(r > 0\)[/tex], multiplying both sides of [tex]\((m - n) < (p + q)\)[/tex] by [tex]\(r\)[/tex] gives us [tex]\((m - n)r < (p + q)r\)[/tex].
### vi) If [tex]\(q + r = 15\)[/tex], then [tex]\(15 = q + r\)[/tex]
This equation employs the Symmetric Property of Equality. The symmetric property of equality states that if [tex]\(a = b\)[/tex], then [tex]\(b = a\)[/tex]. Here, if [tex]\(q + r = 15\)[/tex], then it can also be stated that [tex]\(15 = q + r\)[/tex].
Thus, the properties justifying each statement are:
1. Commutative Property of Addition
2. Distributive Property
3. Multiplicative Identity Property
4. Transitive Property of Inequality
5. Multiplication Property of Inequality
6. Symmetric Property of Equality
