Question

Hessa solved an inequality and graphed the solution on the number line. Which of the following inequalities did she solve?
2/5x−4/5≥ 1 1/5
3/7x+1/7≤ 1 6/7
1/5x−3/5≤ 2/5
3/5x+2/5≥ 3 2/5

a closed circle on the number 5 and an arrow pointing to the left

Answer 1

Solve the inequality 9.5x−1.25<32 . Which of the following graphs shows the solution?

Answer 2

To determine which inequality Hessa solved, we need to analyze the possible solutions and their graphs based on the information provided: she has a closed circle on the number 5 with an arrow pointing to the left. This indicates that the solution set includes 5 and all numbers less than 5.

Let's solve each inequality to check:

1. **\( \frac{2}{5}x - \frac{4}{5} \geq 1 \frac{1}{5} \)**:
\[
\frac{2}{5}x - \frac{4}{5} \geq \frac{6}{5}
\]
Adding \( \frac{4}{5} \) to both sides:
\[
\frac{2}{5}x \geq \frac{6}{5} + \frac{4}{5} \implies \frac{2}{5}x \geq \frac{10}{5} = 2
\]
Multiplying by \( \frac{5}{2} \):
\[
x \geq 5
\]
(This results in an open circle on 5 and an arrow pointing to the right.)

2. **\( \frac{3}{7}x + \frac{1}{7} \leq 1 \frac{6}{7} \)**:
\[
\frac{3}{7}x + \frac{1}{7} \leq \frac{13}{7}
\]
Subtracting \( \frac{1}{7} \):
\[
\frac{3}{7}x \leq \frac{12}{7}
\]
Multiplying by \( \frac{7}{3} \):
\[
x \leq 4
\]
(This results in a closed circle on 4 with an arrow pointing to the left, not a match.)

3. **\( \frac{1}{5}x - \frac{3}{5} \leq \frac{2}{5} \)**:
\[
\frac{1}{5}x - \frac{3}{5} \leq \frac{2}{5}
\]
Adding \( \frac{3}{5} \):
\[
\frac{1}{5}x \leq 1
\]
Multiplying by \( 5 \):
\[
x \leq 5
\]
(This results in a closed circle on 5 with an arrow pointing to the left, which is a match.)

4. **\( \frac{3}{5}x + \frac{2}{5} \geq 3 \frac{2}{5} \)**:
\[
\frac{3}{5}x + \frac{2}{5} \geq \frac{17}{5}
\]
Subtracting \( \frac{2}{5} \):
\[
\frac{3}{5}x \geq 3
\]
Multiplying by \( \frac{5}{3} \):
\[
x \geq 5
\]
(This results in an open circle on 5 and an arrow pointing to the right.)

Based on the analysis, the correct inequality that Hessa solved is:

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

Answer 3

To solve the inequality \( 9.5x - 1.25 < 32 \), follow these steps:

1. **Add \( 1.25 \) to both sides:**
\[
9.5x < 32 + 1.25
\]
\[
9.5x < 33.25
\]

2. **Now, divide both sides by \( 9.5 \):**
\[
x < \frac{33.25}{9.5}
\]

Calculating \( \frac{33.25}{9.5} \):
\[
\frac{33.25}{9.5} = 3.5
\]

So, the solution to the inequality is:
\[
x < 3.5
\]

### Graphing the Solution
1. Since the inequality is \( x < 3.5 \), we will have an **open circle** at **3.5** on the number line (indicating that 3.5 is not included in the solution).
2. The arrow will point to the left, indicating all values less than 3.5.

### Conclusion
Look for a graph with:
- An open circle at **3.5**
- An arrow pointing to the left.

This graph represents the solution to the inequality \( 9.5x - 1.25 < 32 \).