Sure, let's go through these questions step-by-step.
### 7. Write five rational numbers equivalent to [tex]\(\frac{5}{-11}\)[/tex].
To find equivalent fractions, you can multiply the numerator and the denominator of the fraction by the same non-zero integer.
Given fraction: [tex]\(\frac{5}{-11}\)[/tex]
1. Multiply by [tex]\(1\)[/tex]:
[tex]\[
\frac{5 \times 1}{-11 \times 1} = \frac{5}{-11}
\][/tex]
2. Multiply by [tex]\(-1\)[/tex]:
[tex]\[
\frac{5 \times -1}{-11 \times -1} = \frac{-5}{11}
\][/tex]
3. Multiply by [tex]\(2\)[/tex]:
[tex]\[
\frac{5 \times 2}{-11 \times 2} = \frac{10}{-22}
\][/tex]
4. Multiply by [tex]\(-2\)[/tex]:
[tex]\[
\frac{5 \times -2}{-11 \times -2} = \frac{-10}{22}
\][/tex]
5. Multiply by [tex]\(3\)[/tex]:
[tex]\[
\frac{5 \times 3}{-11 \times 3} = \frac{15}{-33}
\][/tex]
Thus, five rational numbers equivalent to [tex]\(\frac{5}{-11}\)[/tex] are:
[tex]\[
\frac{5}{-11}, \quad \frac{-5}{11}, \quad \frac{10}{-22}, \quad \frac{-10}{22}, \quad \frac{15}{-33}
\][/tex]
### 8. Show that [tex]\(\frac{-18}{18}\)[/tex] and [tex]\(\frac{-3}{3}\)[/tex] are equivalent rational numbers.
First, simplify the fractions:
[tex]\(\frac{-18}{18}\)[/tex]:
[tex]\[
-18 \div 18 = -1
\][/tex]
So,
[tex]\[
\frac{-18}{18} = -1
\][/tex]
[tex]\(\frac{-3}{3}\)[/tex]:
[tex]\[
-3 \div 3 = -1
\][/tex]
So,
[tex]\[
\frac{-3}{3} = -1
\][/tex]
Since both [tex]\(\frac{-18}{18}\)[/tex] and [tex]\(\frac{-3}{3}\)[/tex] simplify to [tex]\(-1\)[/tex], we can conclude that:
[tex]\[
\frac{-18}{18} = \frac{-3}{3}
\][/tex]
Thus, [tex]\(\frac{-18}{18}\)[/tex] and [tex]\(\frac{-3}{3}\)[/tex] are equivalent rational numbers.
