Let's address each of your questions one by one.
1. Is subtraction associative in rational numbers? Explain.
To determine if subtraction is associative in rational numbers, we need to check whether
[tex]\[ (a - b) - c = a - (b - c) \][/tex]
for any rational numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
Let's take some examples to understand this.
Suppose [tex]\( a = \frac{5}{3} \)[/tex], [tex]\( b = \frac{2}{3} \)[/tex], and [tex]\( c = \frac{1}{3} \)[/tex].
Calculate [tex]\((a - b) - c\)[/tex]:
[tex]\[ ( \frac{5}{3} - \frac{2}{3} ) - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \][/tex]
Now, calculate [tex]\(a - (b - c)\)[/tex]:
[tex]\[ \frac{5}{3} - ( \frac{2}{3} - \frac{1}{3} ) = \frac{5}{3} - \frac{1}{3} = \frac{5}{3} - \frac{1}{3} = \frac{4}{3} \][/tex]
Since [tex]\(\frac{2}{3} \neq \frac{4}{3}\)[/tex], subtraction is not associative in rational numbers because the grouping changes the result.
2. Verify [tex]\( -(-x) = x \)[/tex] for:
(i) [tex]\( x = \frac{2}{15} \)[/tex]
Let's plug [tex]\( x \)[/tex] into the expression [tex]\( -(-x) \)[/tex]:
[tex]\[ -\left( - \frac{2}{15} \right) = \frac{2}{15} \][/tex]
Thus, [tex]\( -(-x) = x \)[/tex].
(ii) [tex]\( x = \frac{-13}{17} \)[/tex]
Again, plug [tex]\( x \)[/tex] into the expression [tex]\( -(-x) \)[/tex]:
[tex]\[ -\left( - \frac{-13}{17} \right) = -\left( \frac{13}{17} \right) = \frac{-13}{17} \][/tex]
Thus, [tex]\( -(-x) = x \)[/tex].
3. Write -
(i) The set of numbers which do not have an additive inverse.
Every rational number has an additive inverse. For a rational number [tex]\(a\)[/tex], there exists another rational number [tex]\( -a \)[/tex] such that their sum is zero:
[tex]\[ a + (-a) = 0 \][/tex]
Thus, there is no set of numbers which do not have an additive inverse in the context of rational numbers.
Additional Detailed Step-by-Step Solution for the Initial Question
The given equation to solve is:
[tex]\[ 4(18 - 3x) - 9(x + 1) = 0 \][/tex]
First, distribute the constants inside the parentheses:
[tex]\[ 4 \cdot 18 - 4 \cdot 3x - 9 \cdot x - 9 \cdot 1 = 0 \][/tex]
[tex]\[ 72 - 12x - 9x - 9 = 0 \][/tex]
Combine like terms:
[tex]\[ 72 - 21x - 9 = 0 \][/tex]
[tex]\[ 63 - 21x = 0 \][/tex]
Isolate the variable term by moving [tex]\(63\)[/tex] to the other side:
[tex]\[ -21x = -63 \][/tex]
Divide by [tex]\(-21\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{-63}{-21} \][/tex]
[tex]\[ x = 3 \][/tex]
Therefore, the solution to the equation is:
[tex]\[ x = 3 \][/tex]
