Question

2. Find all rational numbers whose absolute values are given below

a. 3.5

4/7

11/6

3 2/5

3. Evaluate each of the following expressions.

a. |- 5| + |5|

b |- 13| - |- 8| + |7|

c |0| + 2 1/3

d |- 8 + 5|

|- 1/13| + |- 15| - 15

4. Evaluate each of the following expressions for the given values of x and y.

a. 5x - |x - 3|, x = - 5

b. |x| - x + 9, x = 3

c. |x + y| - |x|, x = - 3an * dy = 6

d. |x| + |y| x = 5 and y = - 10

e - 3|x + 6| x = - 5

f (|x| - |5y|)/|x + y| x = 4 and y = 8

Answer 1

Let's break down your questions step by step.

### 1. Finding All Rational Numbers with Given Absolute Values

To find all rational numbers whose absolute values are given below, we take both the positive and negative forms of these absolute values:

a. **3.5**
- The rational numbers are: \(3.5\) and \(-3.5\).

b. **\(\frac{4}{7}\)**
- The rational numbers are: \(\frac{4}{7}\) and \(-\frac{4}{7}\).

c. **\(\frac{11}{6}\)**
- The rational numbers are: \(\frac{11}{6}\) and \(-\frac{11}{6}\).

d. **\(3 \frac{2}{5}\)**
- First convert to an improper fraction: \(3 \frac{2}{5} = \frac{15}{5} + \frac{2}{5} = \frac{17}{5}\).
- The rational numbers are: \(\frac{17}{5}\) and \(-\frac{17}{5}\).

### 2. Evaluating Each Expression

a. \(|-5| + |5|\)
- \(|-5| = 5\) and \(|5| = 5\), so \(5 + 5 = 10\).

b. \(|-13| - |-8| + |7|\)
- \(|-13| = 13\), \(|-8| = 8\), and \(|7| = 7\), so \(13 - 8 + 7 = 12\).

c. \(|0| + 2 \frac{1}{3}\)
- \(|0| = 0\), so \(0 + 2 \frac{1}{3} = 2 \frac{1}{3}\) or \(\frac{7}{3}\).

d. \(|-8 + 5|\)
- \(-8 + 5 = -3\), so \(|-3| = 3\).

e. \(|- \frac{1}{13}| + |-15| - 15\)
- \(|- \frac{1}{13}| = \frac{1}{13}\) and \(|-15| = 15\), so \(\frac{1}{13} + 15 - 15 = \frac{1}{13}\).

### 3. Evaluating Expressions for Given Values of x and y

a. \(5x - |x - 3|, x = - 5\)
- \(5(-5) - |-5 - 3| = -25 - |-8| = -25 - 8 = -33\).

b. \(|x| - x + 9, x = 3\)
- \(|3| - 3 + 9 = 3 - 3 + 9 = 9\).

c. \(|x + y| - |x|, x = -3\) and \(y = 6\)
- \(|-3 + 6| - |-3| = |3| - 3 = 3 - 3 = 0\).

d. \(|x| + |y|, x = 5\) and \(y = -10\)
- \(|5| + |-10| = 5 + 10 = 15\).

e. \(-3|x + 6|, x = -5\)
- \(-3|-5 + 6| = -3|1| = -3(1) = -3\).

f. \(\frac{|x| - |5y|}{|x + y|}, x = 4\) and \(y = 8\)
- \(\frac{|4| - |5(8)|}{|4 + 8|} = \frac{4 - 40}{|12|} = \frac{-36}{12} = -3\).

### Summary of Results
- Absolute values: \( 3.5 \to \{3.5, -3.5\}, \frac{4}{7} \to \{\frac{4}{7}, -\frac{4}{7}\}, \frac{11}{6} \to \{\frac{11}{6}, -\frac{11}{6}\}, 3\frac{2}{5} \to \{\frac{17}{5}, -\frac{17}{5}\} \)
- Evaluated expressions:
- a. 10
- b. 12
- c. \(\frac{7}{3}\)
- d. 3
- e. \(\frac{1}{13}\)
- Evaluated expressions for x and y:
- a. -33
- b. 9
- c. 0
- d. 15
- e. -3
- f. -3