Question

On a number line you have a solid dot on 5 then moves back to -10 which of these problems would did you solve
2/4x -4/5 >= 1 1/5
3/5x + 2/5>= 3 2/5
3/7x + 1/7 <= 1 6/7
1/5x - 3/5 <=2/5

Answer 1

Since the line has a solid dot on the 5 does not the 5 have to be included

Answer 2

To determine which inequalities are solved based on the movement from 5 to -10 on a number line, we need to analyze each inequality you provided. We'll solve each one for \( x \) and determine if the solution includes the range from 5 down to -10.

1. **Inequality: \( \frac{2}{4}x - \frac{4}{5} \geq 1 \frac{1}{5} \)**

Simplifying the inequality:
\[
\frac{1}{2}x - \frac{4}{5} \geq \frac{6}{5}
\]

Adding \(\frac{4}{5}\) to both sides:
\[
\frac{1}{2}x \geq \frac{6}{5} + \frac{4}{5}
\]
\[
\frac{1}{2}x \geq \frac{10}{5}
\]
\[
\frac{1}{2}x \geq 2
\]

Multiplying both sides by 2:
\[
x \geq 4
\]
**Solution Range: \( x \geq 4 \)**

2. **Inequality: \( \frac{3}{5}x + \frac{2}{5} \geq 3 \frac{2}{5} \)**

Simplifying the inequality:
\[
\frac{3}{5}x + \frac{2}{5} \geq \frac{17}{5}
\]

Subtracting \(\frac{2}{5}\) from both sides:
\[
\frac{3}{5}x \geq \frac{17}{5} - \frac{2}{5}
\]
\[
\frac{3}{5}x \geq \frac{15}{5}
\]
\[
\frac{3}{5}x \geq 3
\]

Multiplying both sides by \(\frac{5}{3}\):
\[
x \geq 5
\]
**Solution Range: \( x \geq 5 \)**

3. **Inequality: \( \frac{3}{7}x + \frac{1}{7} \leq 1 \frac{6}{7} \)**

Simplifying the inequality:
\[
\frac{3}{7}x + \frac{1}{7} \leq \frac{13}{7}
\]

Subtracting \(\frac{1}{7}\) from both sides:
\[
\frac{3}{7}x \leq \frac{13}{7} - \frac{1}{7}
\]
\[
\frac{3}{7}x \leq \frac{12}{7}
\]

Multiplying both sides by \(\frac{7}{3}\):
\[
x \leq 4
\]
**Solution Range: \( x \leq 4 \)**

4. **Inequality: \( \frac{1}{5}x - \frac{3}{5} \leq \frac{2}{5} \)**

Simplifying the inequality:
\[
\frac{1}{5}x \leq \frac{2}{5} + \frac{3}{5}
\]
\[
\frac{1}{5}x \leq 1
\]

Multiplying both sides by 5:
\[
x \leq 5
\]
**Solution Range: \( x \leq 5 \)**

### Summary of Results:
- **Inequality 1**: \( x \geq 4 \)
- **Inequality 2**: \( x \geq 5 \)
- **Inequality 3**: \( x \leq 4 \) (Fits the move from 5 to -10)
- **Inequality 4**: \( x \leq 5 \)

Given the original movement from 5 to -10, the inequality satisfied during that movement would be **Inequality 3**, which has the solution range \( x \leq 4 \). This means, indeed, you could have solved that inequality during the transition.

Answer 3

Yes, you are correct. When there is a solid dot at 5 on a number line, it indicates that 5 is included in the solution set.

Let's reassess the inequalities that include 5 based on your condition that you moved from 5 to -10:

1. **Inequality: \( \frac{2}{4}x - \frac{4}{5} \geq 1 \frac{1}{5} \)**
- Solution: \( x \geq 4 \) (5 is included)

2. **Inequality: \( \frac{3}{5}x + \frac{2}{5} \geq 3 \frac{2}{5} \)**
- Solution: \( x \geq 5 \) (5 is included)

3. **Inequality: \( \frac{3}{7}x + \frac{1}{7} \leq 1 \frac{6}{7} \)**
- Solution: \( x \leq 4 \) (5 is **not** included)

4. **Inequality: \( \frac{1}{5}x - \frac{3}{5} \leq \frac{2}{5} \)**
- Solution: \( x \leq 5 \) (5 is included)

### Summary with Respect to Inclusion of 5:
- **Inequality 1**: \( x \geq 4 \) (5 is included)
- **Inequality 2**: \( x \geq 5 \) (5 is included)
- **Inequality 3**: \( x \leq 4 \) (5 is **not** included)
- **Inequality 4**: \( x \leq 5 \) (5 is included)

### Final Determination:
The inequalities that include 5 as part of their solution set are:
- **Inequality 1:** \( x \geq 4 \)
- **Inequality 2:** \( x \geq 5 \)
- **Inequality 4:** \( x \leq 5 \)

Since you started at 5 (inclusive) and moved to -10, the inequalities where \( x \) includes 5 are **Inequality 1, Inequality 2, and Inequality 4**.

Thank you for pointing this out!